An arbitrage guide to financial markets

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____________ An Arbitrage Guide to Financial Markets ____________

Robert Dubil

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Wiley Finance Series

Hedge Funds: Quantitative InsightsFrancois-Serge Lhabitant

A Currency Options PrimerShani Shamah

New Risk Measures in Investment and RegulationGiorgio Szego (Editor)

Modelling Prices in Competitive Electricity MarketsDerek Bunn (Editor)

Inflation-indexed Securities: Bonds, Swaps and Other Derivatives, 2nd EditionMark Deacon, Andrew Derry and Dariush Mirfendereski

European Fixed Income Markets: Money, Bond and Interest RatesJonathan Batten, Thomas Fetherston and Peter Szilagyi (Editors)

Global Securitisation and CDOsJohn Deacon

Applied Quantitative Methods for Trading and InvestmentChristian L. Dunis, Jason Laws and Patrick Naim (Editors)

Country Risk Assessment: A Guide to Global Investment StrategyMichel Henry Bouchet, Ephraim Clark and Bertrand Groslambert

Credit Derivatives Pricing Models: Models, Pricing and ImplementationPhilipp J. Schonbucher

Hedge Funds: A Resource for InvestorsSimone Borla

A Foreign Exchange PrimerShani Shamah

The Simple Rules: Revisiting the Art of Financial Risk ManagementErik Banks

Option TheoryPeter James

Risk-adjusted Lending ConditionsWerner Rosenberger

Measuring Market RiskKevin Dowd

An Introduction to Market Risk ManagementKevin Dowd

Behavioural FinanceJames Montier

Asset Management: Equities DemystifiedShanta Acharya

An Introduction to Capital Markets: Products, Strategies, ParticipantsAndrew M. Chisholm

Hedge Funds: Myths and LimitsFrancois-Serge Lhabitant

The Manager’s Concise Guide to RiskJihad S. Nader

Securities Operations: A Guide to Trade and Position ManagementMichael Simmons

Modeling, Measuring and Hedging Operational RiskMarcelo Cruz

Monte Carlo Methods in FinancePeter Jackel

Building and Using Dynamic Interest Rate ModelsKen Kortanek and Vladimir Medvedev

Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured NotesHarry Kat

Advanced Modelling in Finance Using Excel and VBAMary Jackson and Mike Staunton

Operational Risk: Measurement and ModellingJack King

Interest Rate ModellingJessica James and Nick Webber

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Robert Dubil

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Dubil, Robert.An arbitrage guide to financial markets / Robert Dubil.p. cm.—(Wiley finance series)Includes bibliographical references and indexes.ISBN 0-470-85332-8 (cloth : alk. paper)1. Investments—Mathematics.. 2. Arbitrage. 3. Risk. I. Title. II. Series.HG4515.3.D8 2004332.6—dc22 2004010303

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____________________________________________________________________________________________________________________________________________ Contents ____________________________________________________________________________________________________________________________________________

1 The Purpose and Structure of Financial Markets 1

1.1 Overview 11.2 Risk sharing 21.3 The structure of financial markets 81.4 Arbitrage: Pure vs. relative value 121.5 Financial institutions: Asset transformers and broker-dealers 161.6 Primary and secondary markets 181.7 Market players: Hedgers vs. speculators 201.8 Preview of the book 22

Part One SPOT 25

2 Financial Math I—Spot 27

2.1 Interest-rate basics 28Present value 28Compounding 29Day-count conventions 30Rates vs. yields 31

2.2 Zero, coupon and amortizing rates 32Zero-coupon rates 32Coupon rates 33Yield to maturity 35Amortizing rates 38Floating-rate bonds 39

2.3 The term structure of interest rates 40Discounting coupon cash flows with zero rates 42Constructing the zero curve by bootstrapping 44

2.4 Interest-rate risk 49Duration 51Portfolio duration 56

Convexity 57Other risk measures 58

2.5 Equity markets math 58A dividend discount model 60Beware of P/E ratios 63

2.6 Currency markets 64

3 Fixed Income Securities 67

3.1 Money markets 67U.S. Treasury bills 68Federal agency discount notes 69Short-term munis 69Fed Funds (U.S.) and bank overnight refinancing (Europe) 70Repos (RPs) 71Eurodollars and Eurocurrencies 72Negotiable CDs 74Bankers’ acceptances (BAs) 74Commercial paper (CP) 74

3.2 Capital markets: Bonds 79U.S. government and agency bonds 83Government bonds in Europe and Asia 86Corporates 87Munis 88

3.3 Interest-rate swaps 903.4 Mortgage securities 943.5 Asset-backed securities 96

4 Equities, Currencies, and Commodities 101

4.1 Equity markets 101Secondary markets for individual equities in the U.S. 102Secondary markets for individual equities in Europe and Asia 103Depositary receipts and cross-listing 104Stock market trading mechanics 105Stock indexes 106Exchange-traded funds (ETFs) 107Custom baskets 107The role of secondary equity markets in the economy 108

4.2 Currency markets 1094.3 Commodity markets 111

5 Spot Relative Value Trades 113

5.1 Fixed-income strategies 113Zero-coupon stripping and coupon replication 113Duration-matched trades 116Example: Bullet–barbell 116Example: Twos vs. tens 117

viii Contents

Negative convexity in mortgages 118Spread strategies in corporate bonds 121Example: Corporate spread widening/narrowing trade 121Example: Corporate yield curve trades 123Example: Relative spread trade for high and low grades 124

5.2 Equity portfolio strategies 125Example: A non-diversified portfolio and benchmarking 126Example: Sector plays 128

5.3 Spot currency arbitrage 1295.4 Commodity basis trades 131

Part Two FORWARDS 133

6 Financial Math II—Futures and Forwards 135

6.1 Commodity futures mechanics 1386.2 Interest-rate futures and forwards 141

Overview 141Eurocurrency deposits 142Eurodollar futures 142Certainty equivalence of ED futures 146Forward-rate agreements (FRAs) 147Certainty equivalence of FRAs 149

6.3 Stock index futures 149Locking in a forward price of the index 150Fair value of futures 150Fair value with dividends 152Single stock futures 153

6.4 Currency forwards and futures 154Fair value of currency forwards 155Covered interest-rate parity 156Currency futures 158

6.5 Convenience assets—backwardation and contango 1596.6 Commodity futures 1616.7 Spot–Forward arbitrage in interest rates 162

Synthetic LIBOR forwards 163Synthetic zeros 164Floating-rate bonds 165Synthetic equivalence guaranteed by arbitrage 166

6.8 Constructing the zero curve from forwards 1676.9 Recovering forwards from the yield curve 170

The valuation of a floating-rate bond 171Including repo rates in computing forwards 171

6.10 Energy forwards and futures 173

Contents ix

7 Spot–Forward Arbitrage 175

7.1 Currency arbitrage 1767.2 Stock index arbitrage and program trading 1827.3 Bond futures arbitrage 1877.4 Spot–Forward arbitrage in fixed-income markets 189

Zero–Forward trades 189Coupon–Forward trades 191

7.5 Dynamic hedging with a Euro strip 1937.6 Dynamic duration hedge 197

8 Swap Markets 199

8.1 Swap-driven finance 199Fixed-for-fixed currency swap 200Fixed-for-floating interest-rate swap 203Off-market swaps 205

8.2 The anatomy of swaps as packages of forwards 207Fixed-for-fixed currency swap 208Fixed-for-floating interest-rate swap 209Other swaps 210Swap book running 210

8.3 The pricing and hedging of swaps 2118.4 Swap spread risk 2178.5 Structured finance 218

Inverse floater 219Leveraged inverse floater 220Capped floater 221Callable 221Range 222Index principal swap 222

8.6 Equity swaps 2238.7 Commodity and other swaps 2248.8 Swap market statistics 225

Part Three OPTIONS 231

9 Financial Math III—Options 233

9.1 Call and put payoffs at expiry 2359.2 Composite payoffs at expiry 236

Straddles and strangles 236Spreads and combinations 237Binary options 240

9.3 Option values prior to expiry 2409.4 Options, forwards and risk-sharing 2419.5 Currency options 2429.6 Options on non-price variables 243

x Contents

9.7 Binomial options pricing 244One-step examples 244A multi-step example 251Black–Scholes 256Dividends 257

9.8 Residual risk of options: Volatility 258Implied volatility 260Volatility smiles and skews 261

9.9 Interest-rate options, caps, and floors 264Options on bond prices 265Caps and floors 265Relationship to FRAs and swaps 267An application 268

9.10 Swaptions 269Options to cancel 270Relationship to forward swaps 270

9.11 Exotic options 272Periodic caps 272Constant maturity options (CMT or CMS) 273Digitals and ranges 273Quantos 274

10 Option Arbitrage 275

10.1 Cash-and-carry static arbitrage 275Borrowing against the box 275Index arbitrage with options 277Warrant arbitrage 278

10.2 Running an option book: Volatility arbitrage 279Hedging with options on the same underlying 279Volatility skew 282Options with different maturities 284

10.3 Portfolios of options on different underlyings 284Index volatility vs. individual stocks 285Interest-rate caps and floors 286Caps and swaptions 287Explicit correlation bets 288

10.4 Options spanning asset classes 289Convertible bonds 289Quantos and dual-currency bonds with fixed conversion rates 290Dual-currency callable bonds 291

10.5 Option-adjusted spread (OAS) 29110.6 Insurance 292

Long-dated commodity options 293Options on energy prices 294Options on economic variables 294A final word 294

Contents xi

Appendix CREDIT RISK 295

11 Default Risk (Financial Math IV) and Credit Derivatives 297

11.1 A constant default probability model 29811.2 A credit migration model 30011.3 Alternative models 30111.4 Credit exposure calculations for derivatives 30211.5 Credit derivatives 305

Basics 306Credit default swap 306Total-rate-of-return swap 307Credit-linked note 308Credit spread options 308

11.6 Implicit credit arbitrage plays 310Credit arbitrage with swaps 310Callable bonds 310

11.7 Corporate bond trading 310

Index 313

xii Contents

___________________________________________________________________________________________________________________________________________________________________________ 1 __________________________________________________________________________________________________________________________________________________________________________

The Purpose and Structure of

________________________________________________________________________________________________________ Financial Markets ________________________________________________________________________________________________________

1.1 OVERVIEW

Financial markets play a major role in allocating wealth and excess savings to produc-tive ventures in the global economy. This extremely desirable process takes on variousforms. Commercial banks solicit depositors’ funds in order to lend them out to busi-nesses that invest in manufacturing and services or to home buyers who finance newconstruction or redevelopment. Investment banks bring to market offerings of equityand debt from newly formed or expanding corporations. Governments issue short- andlong-term bonds to finance construction of new roads, schools, and transportation net-works. Investors—bank depositors and securities buyers—supply their funds in order toshift their consumption into the future by earning interest, dividends, and capital gains.The process of transferring savings into investment involves various market

participants: individuals, pension and mutual funds, banks, governments, insurancecompanies, industrial corporations, stock exchanges, over-the-counter dealer networks,and others. All these agents can at different times serve as demanders and suppliers offunds, and as transfer facilitators.Economic theorists design optimal securities and institutions to make the process of

transferring savings into investment most efficient. ‘‘Efficient’’ means to produce thebest outcomes—lowest cost, least disputes, fastest, etc.—from the perspective of secur-ity issuers and investors, as well as for society as a whole. We start this book byaddressing briefly some fundamental questions about today’s financial markets. Whydo we have things like stocks, bonds, or mortgage-backed securities? Are they outcomesof optimal design or happenstance? Do we really need ‘‘greedy’’ investment bankers,securities dealers, or brokers soliciting us by phone to purchase unit trusts or mutualfunds? What role do financial exchanges play in today’s economy? Why do developingnations strive to establish stock exchanges even though often they do not have anystocks to trade on them?Once we have basic answers to these questions, it will not be difficult to see why

almost all the financial markets are organically the same. Like automobiles made byToyota and Volkswagen which all have an engine, four wheels, a radiator, a steeringwheel, etc., all interacting in a predetermined way, all markets, whether for stocks,bonds, commodities, currencies, or any other claims to purchasing power, are builtfrom the same basic elements.All markets have two separate segments: original-issue and resale. These are

characterized by different buyers and sellers, and different intermediaries. Theyperform different timing functions. The first transfers capital from the suppliers offunds (investors) to the demanders of capital (businesses). The second transfers

2 An Arbitrage Guide to Financial Markets

capital from the suppliers of capital (investors) to other suppliers of capital (investors).The original-issue and resale segments are formally referred to as:

. Primary markets (issuer-to-investor transactions with investment banks as inter-mediaries in the securities markets, and banks, insurance companies, and others inthe loan markets).

. Secondary markets (investor-to-investor transactions with broker-dealers and ex-changes as intermediaries in the securities markets, and mostly banks in the loanmarkets).

Secondary markets play a critical role in allowing investors in the primary markets totransfer the risks of their investments to other market participants.

All markets have the originators, or issuers, of the claims traded in them (the originaldemanders of funds) and two distinctive groups of agents operating as investors, orsuppliers of funds. The two groups of funds suppliers have completely divergentmotives. The first group aims to eliminate any undesirable risks of the traded assetsand earn money on repackaging risks, the other actively seeks to take on those risks inexchange for uncertain compensation. The two groups are:

. Hedgers (dealers who aim to offset primary risks, be left with short-term or second-ary risks, and earn spread from dealing).

. Speculators (investors who hold positions for longer periods without simultaneouslyholding positions that offset primary risks).

The claims traded in all financial markets can be delivered in three ways. The first is animmediate exchange of an asset for cash. The second is an agreement on the price to bepaid with the exchange taking place at a predetermined time in the future. The last is adelivery in the future contingent on an outcome of a financial event (e.g., level of stockprice or interest rate), with a fee paid upfront for the right of delivery. The three marketsegments based on the delivery type are:

. Spot or cash markets (immediate delivery).

. Forwards markets (mandatory future delivery or settlement).

. Options markets (contingent future delivery or settlement).

We focus on these structural distinctions to bring out the fact that all markets not onlytransfer funds from suppliers to users, but also risk from users to suppliers. They allowrisk transfer or risk sharing between investors. The majority of the trading activity intoday’s market is motivated by risk transfer with the acquirer of risk receiving someform of sure or contingent compensation. The relative price of risk in the market isgoverned by a web of relatively simple arbitrage relationships that link all the markets.These allow market participants to assess instantaneously the relative attractiveness ofvarious investments within each market segment or across all of them. Understandingthese relationships is mandatory for anyone trying to make sense of the vast andcomplex web of today’s markets.

1.2 RISK SHARING

All financial contracts, whether in the form of securities or not, can be viewed asbundles, or packages of unit payoff claims (mini-contracts), each for a specific datein the future and a specific set of outcomes. In financial economics, these are referred toas state-contingent claims.

Let us start with the simplest illustration: an insurance contract. A 1-year life insur-ance policy promising to pay $1,000,000 in the event of the insured’s death can beviewed as a package of 365 daily claims (lottery tickets), each paying $1,000,000 ifthe holder dies on that day. The value of the policy upfront (the premium) is equalto the sum of the values of all the individual tickets. As the holder of the policy goesthrough the year, he can discard tickets that did not pay off, and the value of the policyto him diminishes until it reaches zero at the end of the coverage period.Let us apply the concept of state-contingent claims to known securities. Suppose you

buy one share of XYZ SA stock currently trading at c¼ 45 per share. You intend to holdthe share for 2 years. To simplify things, we assume that the stock trades in incrementsof c¼ 0.05 (tick size). The minimum price is c¼ 0.00 (a limited liability company cannothave a negative value) and the maximum price is c¼ 500.00. The share of XYZ SA can beviewed as a package of claims. Each claim represents a contingent cash flow fromselling the share for a particular price at a particular date and time in the future. Wecan arrange the potential price levels from c¼ 0.00 to c¼ 500.00 in increments of c¼ 0.05 tohave overall 10,001 price levels. We arrange the dates from today to 2 years from today(our holding horizon). Overall we have 730 dates. The stock is equivalent to10,001� 730, or 7,300,730 claims. The easiest way to imagine this set of claims is asa rectangular chessboard where on the horizontal axis we have time and on the verticalthe potential values the stock can take on (states of nature). The price of the stock todayis equal to the sum of the values of all the claims (i.e., all the squares of the chessboard).

The Purpose and Structure of Financial Markets 3

Table 1.1 Stock held for 2 years as a chessboard of contingent claims in two dimensions: time(days 1 through 730) and prices (0.00 through 500.00)

500.00 500.00 500.00 500.00 500.00 500.00 500.00 500.00499.95 499.95 499.95 499.95 499.95 499.95 499.95 499.95499.90 499.90 499.90 499.90 499.90 499.90 499.90 499.90499.85 499.85 499.85 499.85 499.85 499.85 499.85 499.85

. . .60.35 60.35 60.35 60.35 60.35 60.35 60.35 60.3560.30 60.30 60.30 60.30 60.30 60.30 60.30 60.3060.25 60.25 60.25 60.25 60.25 60.25 60.25 60.2560.20 60.20 60.20 60.20 60.20 60.20 60.20 60.2060.15 60.15 60.15 60.15 60.15 60.15 60.15 60.1560.10 60.10 . . . 60.10 60.10 60.10 . . . 60.10 60.10 60.1060.05 60.05 60.05 60.05 60.05 60.05 60.05 60.05 Stock60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 price59.95 59.95 59.95 59.95 59.95 59.95 59.95 59.95 S59.90 59.90 59.90 59.90 59.90 59.90 59.90 59.9059.85 59.85 59.85 59.85 59.85 59.85 59.85 59.8559.80 59.80 59.80 59.80 59.80 59.80 59.80 59.80

. . .0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.450.40 0.40 0.40 0.40 0.40 0.40 0.40 0.400.35 0.35 0.35 0.35 0.35 0.35 0.35 0.350.30 0.30 0.30 0.30 0.30 0.30 0.30 0.300.25 0.25 0.25 0.25 0.25 0.25 0.25 0.250.20 0.20 0.20 0.20 0.20 0.20 0.20 0.200.15 0.15 0.15 0.15 0.15 0.15 0.15 0.150.10 0.10 0.10 0.10 0.10 0.10 0.10 0.100.05 0.05 0.05 0.05 0.05 0.05 0.05 0.050.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1 2 . . . 364 365 366 . . . 729 730

Days

A forward contract on XYZ SA’s stock can be viewed as a subset of this rectangle.Suppose we enter into a contract today to purchase the stock 1 year from today forc¼ 60. We intend to hold the stock for 1 year after that. The forward can be viewed as10,001� 365 rectangle with the first 365 days’ worth of claims taken out (i.e., we are leftwith the latter 365 columns of the board, the first 365 are taken out). The cash flow ofeach claim is equal to the difference between the stock price for that state of nature andthe contract price of c¼ 60. A forward carries an obligation on both sides of the contractso some claims will have a positive value (stock is above c¼ 60) and some negative (stockis below c¼ 60).


Table 1.2 One-year forward buy at c¼ 60 of stock as a chessboard of contingent claims. Payoff incells is equal to S � 60 for year 2. No payoff in year 1

0.00 0.00 0.00 0.00 440.00 440.00 440.00 500.000.00 0.00 0.00 0.00 439.95 439.95 439.95 499.950.00 0.00 0.00 0.00 439.90 439.90 439.90 499.900.00 0.00 0.00 0.00 439.85 439.85 439.85 499.85

0.00 0.00 0.00 0.00 0.35 0.35 0.35 60.350.00 0.00 0.00 0.00 0.30 0.30 0.30 60.300.00 0.00 0.00 0.00 0.25 0.25 0.25 60.250.00 0.00 0.00 0.00 0.20 0.20 0.20 60.200.00 0.00 0.00 0.00 0.15 0.15 0.15 60.150.00 0.00 . . . 0.00 0.00 0.10 . . . 0.10 0.10 60.100.00 0.00 0.00 0.00 0.05 0.05 0.05 60.05 Stock0.00 0.00 0.00 0.00 0.00 0.00 0.00 60.00 price0.00 0.00 0.00 0.00 �0.05 �0.05 �0.05 59.95 S0.00 0.00 0.00 0.00 �0.10 �0.10 �0.10 59.900.00 0.00 0.00 0.00 �0.15 �0.15 �0.15 59.850.00 0.00 0.00 0.00 �0.20 �0.20 �0.20 59.80

0.00 0.00 0.00 0.00 �59.55 �59.55 �59.55 0.450.00 0.00 0.00 0.00 �59.60 �59.60 �59.60 0.400.00 0.00 0.00 0.00 �59.65 �59.65 �59.65 0.350.00 0.00 0.00 0.00 �59.70 . . . �59.70 �59.70 0.300.00 0.00 0.00 0.00 �59.75 �59.75 �59.75 0.250.00 0.00 0.00 0.00 �59.80 �59.80 �59.80 0.200.00 0.00 0.00 0.00 �59.85 �59.85 �59.85 0.150.00 0.00 0.00 0.00 �59.90 �59.90 �59.90 0.100.00 0.00 0.00 0.00 �59.95 �59.95 �59.95 0.050.00 0.00 0.00 0.00 �60.00 �60.00 �60.00 0.00

1 2 . . . 364 365 366 . . . 729 730

Days

An American call option contract to buy XYZ SA’s shares for c¼ 60 with an expiry 2years from today (exercised only if the stock is above c¼ 60) can be represented as a8,800� 730 subset of our original rectangular 10,001� 730 chessboard. This time, thesquares corresponding to the stock prices of c¼ 60 or below are eliminated, because theyhave no value. The payoff of each claim is equal to the intrinsic (exercise) value of thecall. As we will see later, the price of each claim today is equal to at least that.


Table 1.3 American call struck at c¼ 60 as a chessboard of contingent claims. Expiry 2 years.Payoff in cells is equal to S � 60 if S > 60

440.00 440.00 440.00 440.00 440.00 440.00 440.00 500.00439.95 439.95 439.95 439.95 439.95 439.95 439.95 499.95439.90 439.90 439.90 439.90 439.90 439.90 439.90 499.90439.85 439.85 439.85 439.85 439.85 439.85 439.85 499.85

0.35 0.35 0.35 0.35 0.35 0.35 0.35 60.350.30 0.30 0.30 0.30 0.30 0.30 0.30 60.300.25 0.25 0.25 0.25 0.25 0.25 0.25 60.250.20 0.20 0.20 0.20 0.20 0.20 0.20 60.200.15 0.15 0.15 0.15 0.15 0.15 0.15 60.150.10 0.10 . . . 0.10 0.10 0.10 . . . 0.10 0.10 60.100.05 0.05 0.05 0.05 0.05 0.05 0.05 60.05 Stock0.00 0.00 0.00 0.00 0.00 0.00 0.00 60.00 Price0.00 0.00 0.00 0.00 0.00 0.00 0.00 59.95 S0.00 0.00 0.00 0.00 0.00 0.00 0.00 59.900.00 0.00 0.00 0.00 0.00 0.00 0.00 59.850.00 0.00 0.00 0.00 0.00 0.00 0.00 59.80

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.450.00 0.00 0.00 0.00 0.00 0.00 0.00 0.400.00 0.00 0.00 0.00 0.00 . . . 0.00 0.00 0.350.00 0.00 0.00 0.00 0.00 0.00 0.00 0.300.00 0.00 0.00 0.00 0.00 0.00 0.00 0.250.00 0.00 0.00 0.00 0.00 0.00 0.00 0.200.00 0.00 0.00 0.00 0.00 0.00 0.00 0.150.00 0.00 0.00 0.00 0.00 0.00 0.00 0.100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.050.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1 2 . . . 364 365 366 . . . 729 730

Days

Spot securities (Chapters 2–5), forwards (Chapters 6–8), and options (Chapters 9–10)are discussed in detail in subsequent chapters. Here we briefly touch on the valuation ofsecurities and state-contingent claims. The fundamental tenet of the valuation is that ifwe can value each claim (chessboard square) or small sets of claims (entire sections ofthe chessboard) in the package, then we can value the package as a whole. Conversely,if we can value a package, then often we are able to value smaller subsets of claims(through a ‘‘subtraction’’). In addition, we are sometimes able to combine very dis-parate sets of claims (stocks and bonds) to form complex securities (e.g., convertiblebonds). By knowing the value of the combination, we can infer the value of a subset(bullet bond).

In general, the value of a contingent claim does not stay constant over time. If theholder of the life insurance becomes sick during the year and the likelihood of his deathincreases, then likely the value of all claims increases. In the stock example, as informa-tion about the company’s earnings prospects reaches the market, the price of the claimschanges. Not all the claims in the package have to change in value by the same amount.An improvement in the earnings prospects for the company may be only short term.The policyholder’s likelihood of death may increase for all the days immediatelyfollowing his illness, but not for more distant dates. The prices of the individualclaims fluctuate over time, and so does the value of the entire bundle. However, atany given moment of time, given all information available as of that moment, the sumof the values of the claims must be equal to the value of the package, the insurancepolicy, or the stock. We always restrict the valuation effort to here and now, knowingthat we will have to repeat the exercise an instant later.

Let us fix the time to see what assumptions we can make about some of the claims inthe package. In the insurance policy example, we may surmise that the value of theclaims for far-out dates is greater than that for near dates, given that the patient is aliveand well now, and, barring an accident, he is relatively more likely to take time todevelop a life-threatening condition. In the stock example, we assigned the value of c¼ 0to all claims in states with stock exceeding c¼ 500 over the next 2 years, as the likelihoodof reaching these price levels is almost zero. We often assign the value of zero to claimsfor far dates (e.g., beyond 100 years), since the present value of those payoffs, even ifthey are large, is close to zero. We reduce a numerically infinite problem to a finite one.We cap the potential states under consideration, future dates, and times.

A good valuation model has to strive to make the values of the claims in a packageindependent of each other. In our life insurance policy example, the payoff depends onthe person dying on that day and not on whether the person is dead or alive on a givenday. In that setup, only one claim out of the whole set will pay. If we modeled thepayoff to depend on being dead and not dying, all the claims after the morbid eventdate would have positive prices and would be contingent on each other. Sometimes,however, even with the best of efforts, it may be impossible to model the claims in apackage as independent. If a payoff at a later date depends on whether the stockreached some level at an earlier date, the later claim’s value depends on the priorone. A mortgage bond’s payoff at a later date depends on whether the mortgage hasnot already been prepaid. This is referred to as a survival or path-dependence problem.Our imaginary, two-dimensional chessboards cannot handle path dependence and


we ignore this dimension of risk throughout the book as it adds very little to ourdiscussion.Let us turn to the definition of risk sharing:

Definition Risk sharing is a sale, explicit or through a side contract, of all or some ofthe state-contingent claims in the package to another party.

In real life, risk sharing takes on many forms. The owner of the XYZ share may decideto sell a covered call on the stock (see Chapter 10). If he sells an American-style callstruck at c¼ 60 with an expiry date of 2 years from today, he gives the buyer the right topurchase the share at c¼ 60 from him even if XYZ trades higher in the market (e.g., atc¼ 75). The covered call seller is choosing to cap his potential payoff from the stock atc¼ 60 in exchange for an upfront fee (option premium) he receives. This is the same asexchanging the squares corresponding to price levels above c¼ 60 (with values betweenc¼ 60 and c¼ 500) for squares with a flat payoff of c¼ 60.


Table 1.4 Stock plus short American call struck at c¼ 60 as a chessboard of contingent claims.Payoff in cells is equal to 60 if S > 60 and to S if S < 60

60.00 60.00 60.00 60.00 60.00 60.00 60.00 500.0060.00 60.00 60.00 60.00 60.00 60.00 60.00 499.9560.00 60.00 60.00 60.00 60.00 60.00 60.00 499.9060.00 60.00 60.00 60.00 60.00 60.00 60.00 499.85

. . .

60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.3560.00 60.00 60.00 60.00 60.00 60.00 60.00 60.3060.00 60.00 60.00 60.00 60.00 60.00 60.00 60.2560.00 60.00 60.00 60.00 60.00 60.00 60.00 60.2060.00 60.00 . . . 60.00 60.00 60.00 . . . 60.00 60.00 60.1560.00 60.00 60.00 60.00 60.00 60.00 60.00 60.05 Stock60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 price59.95 59.95 59.95 59.95 59.95 59.95 59.95 59.95 S59.90 59.90 59.90 59.90 59.90 59.90 59.90 59.9059.85 59.85 59.85 59.85 59.85 59.85 59.85 59.8559.80 59.80 59.80 59.80 59.80 59.80 59.80 59.80

. . . . . . . . .

0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.450.40 0.40 0.40 0.40 0.40 0.40 0.40 0.400.35 0.35 0.35 0.35 0.35 0.35 0.35 0.350.30 0.30 0.30 0.30 0.30 . . . 0.30 0.30 0.300.25 0.25 0.25 0.25 0.25 0.25 0.25 0.250.20 0.20 0.20 0.20 0.20 0.20 0.20 0.200.15 0.15 0.15 0.15 0.15 0.15 0.15 0.150.10 0.10 0.10 0.10 0.10 0.10 0.10 0.100.05 0.05 0.05 0.05 0.05 0.05 0.05 0.050.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1 2 . . . 364 365 366 . . . 729 730

Days

Another example of risk sharing can be a hedge of a corporate bond with a risk-freegovernment bond. A hedge is a sale of a package of state-contingent claims against aprimary position which eliminates all the essential risk of that position. Only a sale of asecurity that is identical in all aspects to the primary position can eliminate all the risk.A hedge always leaves some risk unhedged ! Let us examine a very common hedge of acorporate with a government bond. An institutional trader purchases a 10-year 5%coupon bond issued by XYZ Corp. In an effort to eliminate interest rate risk, the tradersimultaneously shorts a 10-year 4.5% coupon government bond. The size of the shortis duration-matched to the principal amount of the corporate bond. As Chapter 5explains, this guarantees that for small parallel movements in the interest rates, thechanges in the values of the two positions are identical but opposite in sign. If interestrates rise, the loss on the corporate bond holding will be offset by the gain on theshorted government bond. If interest rates decline, the gain on the corporate bondwill be offset by the loss on the government bond. The trader, in effect, speculatesthat the credit spread on the corporate bond will decline. Irrespective of whetherinterest rates rise or fall, whenever the XYZ credit spread declines, the trader gainssince the corporate bond’s price goes up more or goes down less than that of thegovernment bond. Whenever the credit standing of XYZ worsens and the spreadrises, the trader suffers a loss. The corporate bond is exposed over time to two dimen-sions of risk: interest rates and corporate spread. Our chessboard representing thecorporate bond becomes a large rectangular cube with time, interest rate, and creditspread as dimensions. The government bond hedge eliminates all potential payoffsalong the interest rate axis, reducing the cube to a plane, with only time and creditspread as dimensions. Practically any hedge position discussed in this book can bethought of in the context of a multi-dimensional cube defined by time and risk axes.The hedge eliminates a dimension or a subspace from the cube.

Interest-ratelevel

Spread

Time

Figure 1.1 Reduction of one risk dimension through a hedge. Corporate hedged with agovernment.

1.3 THE STRUCTURE OF FINANCIAL MARKETS

Most people view financial markets like a Saturday bazaar. Buyers spend their cash toacquire paper claims on future earnings, coupon interest, or insurance payouts. If theybuy good claims, their value goes up and they can sell them for more; if they buy badones, their value goes down and they lose money.


When probed a little more on how markets are structured, most finance and econom-ics professionals provide a seemingly more complete description, adding detail aboutwho buys and sells what and why in each market. The respondent is likely to inform usthat businesses need funds in various forms of equity and debt. They issue stock, lease-and asset-backed bonds, unsecured debentures, sell short-term commercial paper, orrely on bank loans. Issuers get the needed funds in exchange for a promise to payinterest payments or dividends in the future. The legal claims on business assets arepurchased by investors, individual and institutional, who spend cash today to get morecash in the future (i.e., they invest). Securities are also bought and sold by governments,banks, real estate investment trusts, leasing companies, and others. The cash-for-paperexchanges are immediate. Investors who want to leverage themselves can borrow cashto buy more securities, but through that they themselves become issuers of broker orbank loans. Both issuers and investors live and die with the markets. When stock pricesincrease, investors who have bought stocks gain; when stock prices decline, they lose.New investors have to ‘‘buy high’’ when share prices rise, but can ‘‘buy low’’ whenshare prices decline. The decline benefits past issuers who ‘‘sold high’’. The rise hurtsthem since they got little money for the previously sold stock and now have to delivergood earnings. In fixed income markets, when interest rates fall, investors gain as thevalue of debt obligations they hold increases. The issuers suffer as the rates they pay onthe existing obligations are higher than the going cost of money. When interest ratesrise, investors lose as the value of debt obligations they hold decreases. The issuersgain as the rates they pay on the existing obligations are lower than the going cost ofmoney.In this view of the markets, both sides—the issuers and the investors—speculate on

the direction of the markets. In a sense, the word investment is a euphemism forspeculation. The direction of the market given the position held determines whetherthe investment turns out good or bad. Most of the time, current issuers and investorshold opposite positions (long vs. short): when investors gain, issuers lose, and viceversa. Current and new participants may also have opposite interests. When equitiesrise or interest rates fall, existing investors gain and existing issuers lose, but newinvestors suffer and new issuers gain.The investor is exposed to market forces as long as he holds the security. He can

enhance or mitigate his exposure, or risk, by concentrating or diversifying the types ofassets held. An equity investor may hold shares of companies from different industrialsectors. A pension fund may hold some positions in domestic equities and somepositions in domestic and foreign bonds to allocate risk exposure to stocks, interestrates, and currencies. The risk is ‘‘good’’ or ‘‘bad’’ depending on whether the investor islong or short on exposure. An investor who has shorted a stock gains when the shareprice declines. A homeowner with an adjustable mortgage gains when interest ratesdecline (he is short interest rates) as the rate he pays resets lower, while a homeownerwith a fixed mortgage loses as he is ‘‘stuck’’ paying a high rate (he is long interest rates).While this standard description of the financial markets appears to be very compre-

hensive, it is rather like a two-dimensional portrait of a multi-dimensional object. Themissing dimension here is the time of delivery. The standard view focuses exclusively onspot markets. Investors purchase securities from issuers or other investors and pay forthem at the time of the purchase. They modify the risks the purchased investmentsexpose them to by diversifying their portfolios or holding shorts against longs in the


same or similar assets. Most tend to be speculators as the universe of hedge securitiesthey face is fairly limited.

Let us introduce the time of delivery into this picture. That is, let us relax theassumption that all trades (i.e., exchanges of securities for cash) are immediate. Con-sider an equity investor who agrees today to buy a stock for a certain market price, butwill deliver cash and receive the stock 1 year from today. The investor is entering into aforward buy transaction. His risk profile is drastically different from that of a spotbuyer. Like the spot stock buyer, he is exposed to the price of the stock, but hisexposure does not start till 1 year from now. He does not care if the stock drops invalue as long as it recovers by the delivery date. He also does not benefit from thetemporary appreciation of the stock compared with the spot buyer who could sell thestock immediately. In our time-risk chessboard with time and stock price on the axes,the forward buy looks like a spot buy with a subplane demarcated by today and 1 yearfrom today taken out. If we ignore the time value of money, the area above the currentprice line corresponds to ‘‘good’’ risk (i.e., a gain), and the area below to ‘‘bad’’ risk(i.e., a loss). A forward sell would cover the same subplane, but the ‘‘good’’ and the‘‘bad’’ areas would be reversed.

Market participants can buy and sell not just spot but also forward. For the purposeof our discussion, it does not matter if, at the future delivery time, what takes place is anactual exchange of securities for cash or just a mark-to-market settlement in cash (seeChapter 6). If the stock is trading at c¼ 75 in the spot market, whether the parties to aprior c¼ 60 forward transaction exchange cash (c¼ 60) for stock (one share) or simplysettle the difference in value with a payment of c¼ 15 is quite irrelevant, as long as thestock is liquid enough so that it can be sold for c¼ 75 without any loss. Also, for ourpurposes, futures contracts can be treated as identical to forwards, even though theyinvolve a daily settlement regimen and may never result in the physical delivery of theunderlying commodity or stock basket.

Let us now further complicate the standard view of the markets by introducing theconcept of contingent delivery time. A trade, or an exchange of a security for cash,agreed on today is not only delayed into the future, but is also made contingent on afuture event. The simplest example is an insurance contract. The payment of a benefiton a $1,000,000 life insurance policy takes place only on the death of the insuredperson. The amount of the benefit is agreed on and fixed upfront between the policy-holder and the issuing company. It can be increased only if the policyholder paysadditional premium. Hazard insurance (fire, auto, flood) is slightly different from lifein that the amount of the benefit depends on the ‘‘size’’ of the future event. The greaterthe damage is, the greater the payment is. An option contract is very similar to a hazardinsurance policy. The amount of the benefit follows a specific formula that depends onthe value of the underlying financial variable in the future (see Chapters 9–10). Forexample, a put option on the S&P 100 index traded on an exchange in Chicago pays thedifference between the selected strike and the value of the index at some future date,times $100 per point, but only if the index goes down below that strike price level. Thebuyer thus insures himself against the index going down and the more the index goesdown the more benefit he obtains from his put option, just as if he held a fire insurancepolicy. Another example is a cap on an interest rate index that provides the holder witha periodic payment every time the underlying interest rate goes above a certain level.Borrowers use caps to protect themselves against interest rate hikes.


Options are used not only for obtaining protection, which is only one form of risksharing, but also for risk taking (i.e., providing specific risk protection for upfrontcompensation). A bank borrower relying on a revolving credit line with an interestrate defined as some spread over the U.S. prime rate or the 3-month c¼ -LIBOR (Londoninterbank offered rate) rate can sell floors to offset the cost of the borrowing. When theindex rate goes down, he is required to make periodic payments to the floor buyerwhich depends on the magnitude of the interest rate decline. He willingly accepts thatrisk because, when rates go down and he has to make the floor payments, the interest heis charged on the revolving loan also declines. In effect, he fixes his minimum borrowingrate in exchange for an upfront premium receipt.Options are not the only packages of contingent claims traded in today’s markets. In

fact, the feature of contingent delivery is embedded in many commonly traded secur-ities. Buyers of convertible bonds exchange their bonds for shares when interest ratesand/or stock prices are high, making the post-conversion equity value higher than thepresent value of the remaining interest on the unconverted bond. Issuers call theiroutstanding callable bonds when interest rates decline below a level at which thevalue of those bonds is higher than the call price. Adjustable mortgages typicallycontain periodic caps that prevent the interest rate and thus the monthly paymentcharged to the homeowner from changing too rapidly from period to period. Manybonds have credit covenants attached to them which require the issuing company tomaintain certain financial ratios, and non-compliance triggers automatic repayment ordefault. Car lease agreements give the lessees the right to purchase the automobile at theend of the lease period for a pre-specified residual value. Lessees sometimes exercisethose rights when the residual value is sufficiently lower than the market price of thevehicle. In many countries, including the U.S., homeowners with fixed-rate mortgagescan prepay their loans partially or fully at any time without penalty. This feature allowshomeowners to refinance their loans with new ones when interest rates drop by asignificant enough margin. The cash flows from the original fixed-rate loans are thuscontingent on interest rates staying high. Other examples abound.The key to understanding these types of securities is the ability to break them down

into simpler components: spot, forward, and contingent delivery. These componentsmay trade separately in the institutional markets, but they are most likely bundledtogether for retail customers or original (primary market) acquirers. Not uncommonly,they are unbundled and rebundled several times during their lives.

Proposition All financial markets evolve to have three structural components: themarket for spot securities, the market for forwards and futures, and the contingentsecurities market which includes options and other derivatives.

All financial markets eventually evolve to have activity in three areas: spot trading forimmediate delivery, trading with forward delivery, and trading with contingent forwarddelivery. Most of the activity of the last two forms is reserved for large institutionswhich is why most people are unaware of them. Yet their existence is necessary for thesmooth functioning of the spot markets. The trading for forward and contingentforward delivery allows dynamic risk sharing for holders of cash securities who tradein and out of contracts tied to different dates and future uncertain events. This risksharing activity, by signaling the constantly changing price of risk, in turn facilitates theflow of the fundamental information that determines the ‘‘bundled’’ value of the spot


securities. In a way, the spot securities that we are all familiar with are the mostcomplicated ones from the informational content perspective. Their value reflects allavailable information about the financial prospects of the entity that issued them andexpectations about the broad market, and is equal to the sum of the values of all state-contingent claims that can be viewed as informational units. The value of forwards andoption-like contracts is tied to more narrow information subsets. These contracts havean expiry date that is short relative to the underlying security and are tailored to specificdimension of risk. Their existence allows the unbundling of the information containedin the spot security. This function is extremely desirable to holders of cash assets as itoffers them a way to sell off undesirable risks and acquire desirable ones at variouspoints in time. If you own a bond issued by a tobacco company, you may be worriedthat legal proceedings against the company may adversely affect the credit spread andthus the value of the bond you hold. You could sell the bond spot and repurchase itforward with the contract date set far into the future. You could purchase a spread-related option or a put option on the bond, or you could sell calls on the bond. All ofthese activities would allow you to share the risks of the bond with another party totailor the duration of the risk sharing to your needs.

1.4 ARBITRAGE: PURE VS. RELATIVE VALUE

In this section, we introduce the notion of relative value arbitrage which drives thetrading behavior of financial firms irrespective of the market they are engaged in.Relative arbitrage takes the concept of pure arbitrage beyond its technical definitionof riskless profit. In it, all primary market risks are eliminated, but some secondarymarket exposures are deliberately left unhedged.

Arbitrage is defined in most textbooks as riskless, instantaneous profit. It occurswhen the law of one price, which states that the same item cannot sell at two differentprices at the same time, is violated. The same stock cannot trade for one price at oneexchange and for a different price at another unless there are fees, taxes, etc. If it does,traders will buy it on the exchange where it sells for less and sell it on the one where itsells for more. Buying Czech korunas for British pounds cannot be more or lessexpensive than buying dollars for pounds and using dollars to buy korunas. If onecan get more korunas for pounds by buying dollars first, no one will buy korunas forpounds directly. On top of that, anyone with access to both markets will convertpounds into korunas via dollars and sell korunas back directly for pounds to realizean instantaneous and riskless profit. This strategy is a very simple example of purearbitrage in the spot currency markets. More complicated pure arbitrage involvesforward and contingent markets. It can take a static form, where the trade is put onat the outset and liquidated once at a future date (e.g., trading forward rate agreementsagainst spot LIBORs for two different terms, see Chapter 6), or a dynamic one, in whichthe trader commits to a series of steps that eliminate all directional market risks andensures virtually riskless profit on completion of these steps. For example, a bonddealer purchases a callable bond from the issuer, buys a swaption from a third partyto offset the call risk, and delta-hedges the rate risk by shorting some bullet swaps. Heguarantees himself a riskless profit provided that neither the issuer nor the swaptionseller defaults. Later chapters abound in detailed examples of both static and dynamicarbitrage.


Definition Pure arbitrage is defined as generating riskless profit today by statically ordynamically matching current and future obligations to exactly offset each, inclusive ofincurring known financing costs.

Not surprisingly, opportunities for pure arbitrage in today’s ultra-sophisticatedmarkets are limited. Most institutions’ money-making activities rely on the principleof relative value arbitrage. Hedge funds and proprietary trading desks of large financialfirms, commonly referred to as arb desks, employ extensively relative arbitrage tech-niques. Relative value arbitrage consists of a broadly defined hedge in which a closesubstitute for a particular risk dimension of the primary security is found and the law ofone price is applied as if the substitute was a perfect match. Typically, the position inthe substitute is opposite to that in the primary security in order to offset the mostsignificant or unwanted risk inherent in the primary security. Other risks are leftpurposely unhedged, but if the substitute is well chosen, they are controllable (exceptin highly leveraged positions). Like pure arbitrage, relative arbitrage can be both staticand dynamic. Let us consider examples of static relative arbitrage.Suppose you buy $100 million of the 30-year U.S. government bond. At the same

time you sell (short) $102 million of the 26-year bond. The amounts $100 and $102 arechosen through ‘‘duration matching’’ (see Chapters 2 and 5) which ensures that wheninterest rates go up or down by a few basis points the gains on one position exactlyoffset the losses on the other. The only way the combined position makes or losesmoney is when interest rates do not change in parallel (i.e., the 30-year rates changeby more or less than the 26-year rates). The combined position is not risk-free; it isspeculative, but only in a secondary risk factor. Investors hardly distinguish between30- and 26-year rates; they worry about the overall level of rates. The two rates tend tomove closely together. The relative arbitrageur bets that they will diverge.The bulk of swap trading in the world (Chapter 8) relies on static relative arbitrage.

An interest rate swap dealer agrees to pay a fixed coupon stream to a corporate cus-tomer, himself an issuer of a fixed-rate bond. The dealer hedges by buying a fixedcoupon government bond. He eliminates any exposure to interest rate movements ascoupon receipts from the government bond offset the swap payments, but is left withswap spread risk. If the credit quality of the issuer deteriorates, the swap becomes‘‘unfair’’ and the combined position has a negative present value to the dealer.Dynamic relative arbitrage is slightly more complicated in that the hedge must be

rebalanced continuously according to very specific computable rules. A seller of a 3-year over-the-counter (OTC) equity call may hedge by buying 3- and 6-month calls onthe exchange and shorting some of the stock. He then must rebalance the number ofshares he is short on a daily basis as the price of those shares fluctuates. This so-calleddelta hedge (see Chapters 9–10) eliminates exposure to the price risk. The mainunhedged exposure is to the implied volatility differences between the options soldand bought. In the preceding static swap example, the swap dealer may elect not tomatch the cash flows exactly on each swap he enters into; instead, he may take positionsin a small number of ‘‘benchmark’’ bonds in order to offset the cash flows in bulk. Thisshortcut, however, will require him to dynamically rebalance the portfolio of bonds.This book explains the functioning of financial markets by bringing out pure and

relative value arbitrage linkages between different market segments. Our examples


appear complicated as they involve futures, options, and other derivatives, but they allrely on the same simple principle of seeking profit through selective risk elimination.

Definition Relative value arbitrage is defined as generating profit today by statically ordynamically matching current and future obligations to nearly offset each other, net ofincurring closely estimable financing costs.

To an untrained eye, the difference between relative value arbitrage and speculation istenuous; to a professional, the two are easily discernible. A popular equity tradingstrategy called pairs trading (see Chapter 5) is a good case in point. The strategy ofbuying Pfizer (PFE) stock and selling GlaxoSmithKline (GSK) is pure speculation. Onecan argue that both companies are in pharmaceuticals, both are large, and both withsimilar R&D budgets and new drug pipelines. The specific risks of the two companies,however, are quite different and they cannot be considered close substitutes. BuyingPolish zlotys with British pounds and selling Czech korunas for British pounds is alsoan example of speculation, not of relative value arbitrage. Polish zlotys and Czechkorunas are not close substitutes. An in-between case, but clearly on the speculativeside, is called a basis trade. An airline needing to lock in the future prices of jet fuel,instead of entering into a long-term contract with a refiner, buys a series of crude oilfutures, the idea being that supply shocks that cause oil prices to rise affect jet fuel in thesame way. When prices increase, the airline pays higher prices for jet fuel, but profitsfrom oil futures offset those increases, leaving the total cost of acquiring jet fuelunchanged. Buying oil futures is appealing as it allows liquidating the protectionscheme when prices decline instead of rising or getting out halfway through an increase.This trade is not uncommon, but it exposes the airline to the basis risk. When supplyshocks take place at the refinery level not the oil delivery level, spot jet fuel prices mayincrease more rapidly than crude oil futures.

Most derivatives dealers espouse the relative value arbitrage principle. They selloptions and at the same time buy or sell the underlying stocks, bonds, or mortgagesin the right proportions to exactly offset the value changes of the sold option and theposition in the underlying financial asset. Their lives are, however, quite complicated inthat they have to repeat the exercise every day as long as the options they sold are alive,even if they do not sell additional options. This is because the appropriate proportionsof the underlyings they need to buy or sell change every day. These proportions orhedge ratios depend on changing market factors. It is these market factors that are thesecondary risks the dealers are exposed to. The dynamic rebalancing of the positionsserves to create a close substitute to the options sold, but it does not offset all the risks.

Relative value arbitrage in most markets relies on a building block of a static ordynamic cash-and-carry trade. The static version of the cash-and-carry trade (intro-duced in Chapters 6–7) consists typically of a spot purchase (for cash) and a forwardsell, or the reverse. The dynamic trade (introduced in Chapter 9), like in the precedingoption example, consists of a series of spot purchases or sales at different dates and acontingent payoff at the forward date. The glue that ties the spot and the forwardtogether is the cost of financing, or the carry, of the borrowing to buy spot orlending after a spot sale. Even the most complicated structured derivative transactionsare combinations of such building blocks across different markets. When analyzingsuch trades, focusing on institutional and market infrastructure details in each


market can only becloud this basic structure of arbitrage. This book clarifies the essenceof such trades by emphasizing common elements. It also explains why most institutionsrely on the interaction of dealers on large trading floors to take advantage of inter-market arbitrages. The principle of arbitrage is exploited not only to show whatmotivates traders to participate in each market (program trading of stock indexfutures vs. stock baskets, fixed coupon stripping in bonds, triangular arbitrage incurrencies, etc.), but also what drives the risk arbitrage between markets (simultaneoustrades in currencies in money markets, hedging mortgage servicing contracts with swapoptions, etc.).Many readers view no-arbitrage conditions found in finance textbooks as strict mathe-

matical constructs. It should be clear from the above discussion that they are notmathematical at all. These equations do not represent the will of God, like thosepertaining to gravity or thermodynamics in physics. They stem from and are continu-ously ensured by the most basic human characteristic: greed. Dealers tirelessly look todiscover pure and relative value arbitrage (i.e., opportunities to buy something at oneprice and to sell a disguised version of the same thing for another price). By executingtrades to take advantage of the temporary deviations from these paramount rules, theyeliminate them by moving prices back in line where riskless money cannot be made and,by extension, the equations are satisfied.In this book, all the mathematical formulae are traced back to the financial transac-

tions that motivate them. We overemphasize the difference between speculation andpure arbitrage in order to bring out the notion of relative value arbitrage (sometimesalso referred to as risk arbitrage). Apart from the ever-shrinking commissions, mosttraders earn profit from ‘‘spread’’—a reward for relative value risk arbitrage. A swaptrader, who fixes the borrowing rate for a corporate client, hedges by selling Treasurybonds. He engages in a relative value trade (swaps vs. government bonds) whichexposes him to swap spread movements. A bank that borrows by opening new checkingdeposits and lends by issuing mortgages eliminates the risk of parallel interest ratemovements (which perhaps affect deposit and mortgage rates to the same degree),but leaves itself exposed to yield curve tilts (non-parallel movements) or default risk.In all these cases, the largest risks (the exposure to interest rate changes) are hedged out,and the dealer is left exposed to secondary ones (swap spread, default).Most forms of what is conventionally labeled as investment under our definition

qualify as speculation. A stock investor who does not hedge, or risk-share in someway, is exposed to the primary price risk of his asset. It is expected in our lives that,barring short-term fluctuations, over time the value of our assets increases. Theeconomy in general grows, productivity increases, and our incomes rise as we acquiremore experience. We find ourselves having to save for future consumption, family, andretirement. Most of the time, often indirectly through pension and mutual funds, we‘‘invest’’ in real estate, stocks, and bonds. Knowingly or not, we speculate. Financialinstitutions, as their assets grow, find themselves in the same position. Recognizing thatfact, they put their capital to use in new products and services. They speculate on theirsuccess. However, a lot of today’s institutional dealers’ trading activity is not driven bythe desire to bet their institutions’ capital on buy-low/sell-high speculative ventures.Institutional traders do not want to take primary risks by speculating on markets to goup or down; instead, they hedge the primary risks by simultaneously buying and sellingor borrowing and lending in spot, forward, and option markets. They leave themselves


exposed only to secondary ‘‘spread’’ risks. Well-managed financial institutions arecompensated for taking those secondary risks. Even the most apt business schoolstudents often misunderstand this fine distinction between speculation and relativevalue arbitrage. CEOs often do too. Nearly everyone has heard of the Barings, IGMetallgesellschaft, and Orange County fiascos of the 1990s. History is filled withexamples of financial institutions gone bankrupt as a result of gambling.

Institutional trading floors are designed to best take advantage of relative arbitragewithin each market. They are arranged around individual trading desks, surrounded byassociated marketing and clearing teams, each covering customers within a specificmarket segment. Trading desks that are likely to buy each other’s products areplaced next to each other. Special proprietary desks (for short, called prop or arbdesks) deal with many customer desks of the same firm or other firms and manyoutside customers in various markets. Their job is to specifically focus on relativevalue trades or outright speculation across markets. The distinction between the twotypes of desks—customer vs. proprietary—is in constant flux as some markets expandand some shrink. Trading desks may collaborate in the types of transactions theyengage in. For example, a money market desk arranges an issuance of short-termpaper whose coupon depends on a stock index. It then arranges a trade between thecustomer and its swap desk to alter the interest rate exposure profile and between thecustomer and the equity derivatives desk to eliminate the customer’s exposure to equityrisk. The customer ends up with low cost of financing and no equity risks. The dealerfirm lays off the swap and equity risk with another institution. Hundreds of suchintermarket transactions take place every day in the dealing houses in London, NewYork, and Hong Kong.

Commercial banks operate on the same principle. They bundle mortgage, car loan,or credit card receipts into securities with multiple risk characteristics and sell theunwanted ones to other banks. They eliminate the prepayment risk in their mortgageportfolios by buying swaptions from swap dealers.

1.5 FINANCIAL INSTITUTIONS: ASSET TRANSFORMERS

AND BROKER-DEALERS

Financial institutions can be broadly divided into two categories based on their raisond’etre:

. Asset transformers.

. Broker-dealers.

The easiest way to identify them is by examining their balance sheets. Asset transfor-mers’ assets have different legal characteristics from their liabilities. Broker-dealers mayhave different mixes on the two sides of the balance sheet, but the categories tend to bethe same.

An asset transformer is an institution that invests in certain assets, but issues liabil-ities in the form designed to appeal to a particular group of customers. The bestexample is a commercial bank. On the asset side, a bank issues consumer (mortgage,auto) and business loans, invests in bonds, etc. The main form of liability it issues ischecking accounts, saving accounts, and certificates of deposit (CDs). Customers


specifically desire these vehicles as they facilitate their day-to-day transactions and oftenoffer security of government insurance against the bank’s insolvency. For example, inthe U.S. the Federal Deposit Insurance Corporation (FDIC) guarantees all deposits upto $100,000 per customer per bank. The bank’s customers do not want to invest directlyin the bank’s assets. This would be quite inconvenient as they would have to buy andsell these ‘‘bulky’’ assets frequently to meet their normal living expenditures. From aretail customer’s perspective, the bank’s assets often have undesirably long maturitywhich entails price risk if they are sold quickly, and they are offered only in largedenominations. In order to attract funding, the bank repackages its mortgage andbusiness loan assets into liabilities, such as checking accounts and CDs, which havemore palatable characteristics: immediate cash machine access, small denomination,short maturity, and deposit insurance. Another example of an asset transformer is amutual fund (or a unit investment trust). A mutual fund invests in a diversified portfolioof stocks, bonds, or money market instruments, but issues to its customers smalldenomination, easily redeemable participation shares (unit trust certificates) andoffers a variety of services, like daily net asset value calculation, fund redemptionand exchange, or a limited check-writing ability. Other large asset transformers areinsurance companies that invest in real estate, stocks, and bonds (assets), but issuepolicies with payouts tied to life or hardship events (liabilities). Asset transformersare subject to special regulations and government supervision. Banks require bankcharters to operate, are subject to central bank oversight, and must belong to depositinsurance schemes. Mutual fund regulation is aimed at protecting small investors (e.g.,as provided for by the Investment Company Act in the U.S.). Insurance companiesrates are often sanctioned by state insurance boards. The laws in all these cases setspecific forms of legal liabilities asset transformers may create and sound investmentguidelines they must follow (e.g., percentage of assets in a particular category). Assettransformers are compensated largely for their role in repackaging their assets withundesirable features into liabilities with customer-friendly features. That very activityautomatically introduces great risks into their operations. Bank liabilities have muchshorter duration (checking accounts) than their assets (fixed-rate mortgages). If interestrates do not move in parallel, the spread they earn (interest differential between ratescharged on loans and rates paid on deposits) fluctuates and can be negative. Theypursue relative value arbitrage in order to reduce this duration gap.Broker-dealers do not change the legal and functional form of the securities they own

and owe. They buy stocks, currencies, mortgage bonds, leases, etc. and they sell thesame securities. As dealers they own them temporarily before they sell them, exposingthemselves to temporary market risks. As brokers they simply match buyers and sellers.Broker-dealers participate in both primary sale and secondary resale transactions. Theytransfer securities from the original issuers to buyers as well as from existing owners tonew owners. The first is known as investment banking or corporate finance, the latter asdealing or trading. The purest forms of broker-dealers exist in the U.S. and Japan wherelaws have historically separated them from other forms of banking. Most securitiesfirms in those two countries are pure broker-dealers (investment banking, institutionaltrading, and retail brokerage) with an addition of asset-transforming businesses of assetmanagement and lending. In most of continental Europe, financial institutions areconglomerates commonly referred to as universal banks as they combine both functions.In recent years, with the repeal of the Glass–Steagal Act in the U.S. and the wave of


consolidations taking place on both sides of the Atlantic, U.S. firms have the possibilityto converge more closely to the European model. Broker-dealers tend to be much lessregulated than asset transformers and the focus of laws tends to be on small investorprotection (securities disclosure, fiduciary responsibilities of advisers, etc.).

Asset transformers and broker-dealers compete for each other’s business. Securitiesfirms engage in secured and unsecured lending and offer check-writing in their broker-age accounts. They also compete with mutual funds by creating bundled or indexedsecurities designed to offer the same benefits of diversification. In the U.S., the tradingon the American Stock Exchange is dominated by ETFs (exchange-traded funds),HOLDRs (holding company despositary receipts), Cubes, etc., all of which are de-signed to compete with index funds, instead of ordinary shares. Commercial bankssecuritize their credit card and mortgage loans to trade them out of their balancesheets. The overall trend has been toward disintermediation (i.e., securitization of pre-viously transformed assets into more standardized tradeable packages). As burdensomeregulations fall and costs of securitization plummet, retail customers are increasinglygiven access to markets previously reserved for institutions.

1.6 PRIMARY AND SECONDARY MARKETS

From the welfare perspective, the primary role of financial markets has always been totransfer funds between suppliers of excess funds and their users. The users includebusinesses that produce goods and services in the economy, households that demandmortgage and consumer loans, governments that build roads and schools, financialinstitutions, and many others. All of these economic agents are involved in productiveactivities that are deemed economically and socially desirable. Throughout most ofhistory, it was bankers and banks who made that transfer of funds possible by accept-ing funds from depositors and lending them to kings, commercial ventures, and others.With the transition from feudalism to capitalism came the new vehicles of performingthat transfer in the form of shares in limited liability companies and bonds issued bysovereigns and corporations. Stock, bond, and commodity exchanges were formed toallow original investors in these securities to efficiently share the risks of these instru-ments with new investors. This in turn induced many suppliers of funds to becomeinvestors in the first place as the risks of holding ‘‘paper’’ were diminished. ‘‘Paper’’could be easily sold and funds recovered. A specialized class of traders emerged whodealt only with trading ‘‘paper’’ on the exchanges or OTC. To them paper was and isfaceless. At the same time, the old role of finding new productive ventures in need ofcapital has shifted from bankers to investment bankers who, instead of granting loans,specialized in creating new shares and bonds for sale to investors for the first time. Toinvestment bankers, paper is far from faceless. Prior to the launch of any issue, the mainjob of an investment banker or his corporate finance staff, like that of a loan banker, isto evaluate the issuing company’s business, its financial condition and to prepare avaluation analysis for the offered security.

As we stated before, financial markets for securities are organized into two segmentsdefined by the parties to a securities transaction:

. Primary markets.

. Secondary markets.


This segregation exists only in securities, not in private party contracts like OTCderivatives. In private contracts, the primary market issuers also tend to be the second-ary market traders, and the secondary market operates through assignments andmark-to-market settlements rather than through resale.In primary markets, the suppliers of funds transfer their excess funds directly to the

users of funds through a purchase of securities. An investment banker acts as anintermediary, but the paper-for-cash exchange is between the issuing company andthe investor. The shares are sold either publicly, through an initial public offering ora seasoned offering, or privately through a private placement with ‘‘qualified investors’’,typically large institutions. Securities laws of the country in which the shares are soldspell out all the steps the investment bank must take in order to bring the issue tomarket. For example, in the U.S. the shares must be registered with the Securities andExchange Commission (SEC), a prospectus must be presented to new investors prior toa sale, etc. Private placements follow different rules, the presumption being that largequalified investors need less protection than retail investors. In the U.S. they aregoverned by Rule 144-A which allows their subsequent secondary trading through asystem similar to an exchange.In secondary markets, securities are bought and sold only by investors without the

involvement of the original user of funds. Secondary markets can be organized asexchanges or as OTC networks of dealers connected by phone or computer, or ahybrid of the two. The Deutsche Borse and the New York Stock Exchange (NYSE)are examples of organized exchanges. It is worth noting however that exchanges differgreatly from each other. The NYSE gives access to trade flow information to humanmarket-makers called specialists to ensure the continuity of the market-making in agiven stock, while the Tokyo Stock Exchange is an electronic market where continuityis not guaranteed, but no dealer can earn monopoly rents from private informationabout buys and sells (see Chapter 4). Corporate and government bond trading (seeChapter 3) are the best examples of OTC markets. There, like in swap and currencymarkets, all participants are dealers who trade one on one for their own account. Theymaintain contact with each other over a phone and computer network, and jointlypolice the fair conduct rules through industry associations. For example, in the OTCderivatives markets, the International Swap Dealers Association (ISDA) standardizesthe terminology used in quoting the terms and rates, and formalizes the documentationused in confirming trades for a variety of swap and credit derivative agreements. Thebest example of a hybrid between an exchange and an OTC market is the NASDAQ inthe U.S. The exchange is only virtual as participants are connected through a computersystem. Access is limited to members only and all members are dealers.Developing countries strive to create smooth functioning secondary markets. They

often rush to open stock exchanges even though there may only be a handful of com-panies large enough to have a significant number of dispersed shareholders. In order toimprove the liquidity of trading, nascent exchanges limit the number of exchange seatsto very few, the operating hours to sometimes only one per day, etc. All these efforts areaimed at funneling all buyers and sellers into one venue. This parallels the goals of thespecialist system on the NYSE. Developing countries’ governments strive to establish awell-functioning government bond market. They start by issuing short-term obligationsand introduce longer maturities as quickly as the market will have an appetite forthem.


The main objective in establishing these secondary trading places is to lower the costof raising capital in the primary markets by offering the primary market investors alarge outlet for risk sharing. Unless investors are convinced that they can easily get inand out of these securities, they will not buy the equities and bonds offered by theissuers (local businesses and governments) in the first place. This ‘‘tail wag the dog’’pattern of creating secondary markets first is very typical not only for lesser developednations, but is quite common in introducing brand new risk classes into the market-place. In the late 1980s, Michael Milken’s success in selling highly speculative high-yieldbonds to investors relied primarily in creating a secondary OTC market by assuringactive market-making by his firm Drexel Burnham Lambert. Similarly, prior to itscollapse in 2002, Enron’s success in originating energy forwards and contingent con-tracts was driven by Enron’s ability to establish itself as a virtual exchange of energyderivatives (with Enron acting as the monopolist dealer, of course). In both of thesecases, the firms behind the creation of these markets failed, but the primary andsecondary markets they started remained strong, the high-yield market being one ofthe booming high performers during the tech stock bubble collapse in 2000–02.

1.7 MARKET PLAYERS: HEDGERS VS. SPECULATORS

According to a common saying, nothing in life is certain except death and taxes. Noinvestment in the market is riskless, even if it is in some way guaranteed. Let uschallenge some seemingly intuitive notions of what is risky and what is safe.

Sparkasse savers in Germany, postal account holders in Japan, U.S. Treasury Billinvestors, for most intents and purposes avoid default risk and are guaranteed a pos-itive nominal return on their savings. T-Bill and CD investors lock in the rates until thematurity of the instruments they hold. Are they then risk-free investors and not spec-ulators? They can calculate in advance the exact dollar amount their investment willpay at maturity. After subtraction of the original investment, the computed percentagereturn will always be positive. Yet, by locking in the cash flows, they are forgoing thechance to make more. If, while they are holding their CD, short-term or rollover ratesincrease, they will have lost the extra opportunity return they could have earned. We arehinting here at the notion of opportunity cost of capital common in finance.

Let us consider another example. John Smith uses the $1,000 he got from his uncle topurchase shares in XYZ Corp. After 1 year, he sells his shares for $1,100. His annualreturn is 10%. Adam Jones borrows $1,000 at 5% from his broker to purchase shares inXYZ Corp. After 1 year, he sells his shares for $1,100. His annual return is 10% onXYZ shares, but he has to pay 5%, or $50, interest on the loan, so his net return is 5%.Should we praise John for earning 10% on his capital and scold Adam for earning only5%? Obviously not. Adam’s cost of capital was 5%. So was John’s! His was thenebulous opportunity cost of capital, or a shadow cost. He could have earned 5%virtually risk-free by lending to the broker instead of investing in risky shares. So hisrelative return, or excess return, was only 5%. In our T-Bill or CD example, one canargue that an investor in a fixed-rate CD is a speculator as he gambles on the rates notincreasing prior to the maturity of his CD. The fact that his net receipts from the CD atmaturity are guaranteed to be positive is irrelevant. There is nothing special about a 0%threshold for your return objective (especially if one takes into account inflation).


In the context of this book, all investors who take a position in an asset, whether byborrowing or using owned funds, and the asset’s return over its life is not contractuallyidentical to the investor’s cost of capital, will be considered speculators. This definitionis only relative to some benchmark cost of capital. In this sense both Adam and Johnspeculate by acquiring shares whose rate of return differs from their cost of capital of5%. An outright CD investment is speculative as the rate on the CD is not guaranteedto be the same as that obtained by leaving the investment in a variable rate moneymarket account. A homeowner who takes out a fixed-rate mortgage to finance a housepurchase is a speculator even though he fixes his monthly payments for the next 30years! When he refinances his loan, he cancels a prior bet on interest rates and places anew one. In contrast, an adjustable rate mortgage borrower pays the fair market rateevery period equal to the short-term rate plus a fixed margin.Most financial market participants can be divided into two categories based on

whether their capital is used to place bets on the direction of the market prices orrates or whether it is used to finance holdings of sets of transactions which largelyoffset each other’s primary risks: speculators and hedgers.Speculators are economic agents who take on explicit market risks in order to earn

returns in excess of their cost of capital. The risks they are exposed to through theirinvestments are not offset by simultaneous ‘‘hedge’’ transactions. Hedgers are economicagents who enter into simultaneous transactions designed to have offsetting marketrisks in such a way that the net returns they earn are over and above their cost ofcapital. All arbitrageurs, whether pure or relative, are hedgers. They aim to earn nearlyrisk-free returns after paying all their financing costs. A pure arbitrageur’s or stricthedger’s returns are completely risk-free. A relative arbitrageur’s returns are not risk-free; he is exposed to secondary market risks.All ‘‘investors’’ who use their capital to explicitly take on market risks are specula-

tors. Their capital often comes in the form of an outside endowment. Mutual fundsobtain fresh funds by shareholders sending them cash. Pension funds get capital frompayroll deductions. Insurance companies sell life or hazard policies and invest thepremiums in stocks, bonds, and real estate. Individual investors deposit cash intotheir brokerage accounts in order to buy, sell, or short-sell stocks and bonds. In allthese cases, the ‘‘investors’’ use their funds (i.e., sacrifice their cost of capital) to bet onthe direction of the market they invest in. They ‘‘buy’’ the services of brokers anddealers who facilitate their investment strategies. In order to help these investorsimprove the precision of the bets they take, broker-dealers who are hedgers bynature invent new products that they ‘‘sell’’ to investors. These can be new types ofbonds, warrants and other derivatives, new classes of shares, new types of trusts, andannuities. Often, the division of the players into speculators and hedgers is replaced bythe alternative terms of:

. Buy-side participants.

. Sell-side participants.

Buy-side players are investors who do not originate the new investment vehicles. Theychoose from a menu offered to them by the sell-side players. The sell-siders try to avoidgambling their own capital on the explicit direction of the market. They want to usetheir capital to finance the hedge (i.e., to ‘‘manufacture’’ the new products). As soonas they ‘‘sell’’ them, they look to enter into a largely offsetting trade with another


counterparty or to hedge the risks through a relative arbitrage strategy. Often thesell-sider’s hedge strategies are very imperfect and take time to arrange (i.e., whensell-siders act as speculators). The hedger/speculator compartmentalization is notexactly equivalent to sell/buy-side division. Sell-siders often act as both hedgers andspeculators, but their mindset is more like that of the hedger (‘‘to find the other side ofthe trade’’). Buy-siders enter into transactions with sell-siders in order to get exposed toor alter how they are exposed to market risks (‘‘to get in on a trade’’).

We use quotes around the words ‘‘buy’’ and ‘‘sell’’ to emphasize that the sell-siderdoes not necessarily sell a stock or bond to a buy-sider. He can just as well buy it. Buthe hedges his transaction while the buy-sider does not.

Geographically, the sell-side resides in global financial centers, like New York orLondon, and is represented by the largest 50 global financial institutions. The buy-sideis very dispersed and includes all medium and smaller banks with mostly commercialbusiness, all mutual and pension funds, some university endowments, all insurancecompanies, and all finance corporations. The buy/sell and hedger/speculator distinc-tions have recently become blurred. Larger regional banks in the U.S. which havetraditionally been buy-side institutions started their own institutional trading busi-nesses. They now offer security placement and new derivative product services tosmaller banks and thrifts. In the 1990s, some insurance companies established sell-side trading subsidiaries and used their capital strength and credit rating to competevigorously with broker-dealers. Most of these subsidiaries have the phrase ‘‘FinancialProducts’’ inserted in their name (e.g., Gen Re Financial Products or AIG FP).

One type of company that can be by design on both the buy- and sell-side is a hedgefund. Hedge funds are capitalized like typical speculators (read: investment companies),similar to mutual funds, but without the regulatory protection of the small investor. Yetalmost all hedge fund strategies are some form of relative value arbitrage (i.e., they arehedges). The original capital is used only to acquire leverage and to replicate a hedgestrategy as much as possible. Most hedge funds have been traditionally buy-siders.They have tended not to innovate, but to use off-the-shelf contracts from dealers.Sometimes, however, hedge funds grow so large in their market segment that theyare able to wrest control of the demand and supply information flow from thedealers and are able to sell hedges to the dealers, effectively becoming sellers ofinnovative strategies. In the late 1990s, funds like Tiger, AIM, or LTCM, sometimesput on very large hedged positions, crowding dealers only into speculative choices as thesupply of available hedges was exhausted by the funds. The early 2000s have seen thereturn of hedge funds to their more traditional buy-side role as the average size ofthe fund declined and the number of funds increased dramatically.

1.8 PREVIEW OF THE BOOK

Most financial markets textbooks are organized by following markets for different typesof securities: stocks, money markets, bonds, mortgages, asset-backed securities, realestate trusts, currencies, commodities, etc. This is analogous to reviewing the carindustry alphabetically by make, starting with Acura, Audi, and BMW and endingwith Volkswagen and Volvo. This book is arranged structurally to emphasize the


common features of all the segments, analogous to describing the engines first, thenchassis and body, and ending with safety features and interior comforts. This allows thereader to fully understand the internal workings of the markets, rather than learningabout unimportant institutional details. The book is divided into four parts:

. Part I: Spot—trading in cash securities for immediate delivery, arbitrage throughspot buying, and selling of like securities that trade at different prices.

. Part II: Forwards—futures and forward contracts for future delivery of the under-lying cash assets, arbitrage through static cash-and-carry (i.e., spot buy or sell),supplemented by borrow or lend against a forward sell or buy).

. Part III: Options—derivative contracts for contingent future delivery of the under-lying cash assets, arbitrage through dynamic cash-and-carry, or delta-hedge (i.e.,continuous rebalancing of the cash-and-carry position).

. Appendix: Credit Risk—default-risky delivery (spot, forward, or contingent), arbit-rage through all three strategies using default-risky assets against default-free assets.

The first three represent the fundamental building blocks present in any market. Eachpart is defined by the delivery time and form of transactions. Each has its own internalno-arbitrage rules (law of one price) and each is related to the other two by another setof no-arbitrage rules (static and dynamic cash-and-carry equations). As we will show,all the no-arbitrage rules, player motivations, and trade strategies of each segment inmarkets for different securities (stocks, bonds, etc.) are strikingly similar.The appendix of the book is dedicated to credit risks that cut across all three

dimensions of financial markets. This part enjoys the lightest treatment in the bookand contains only one chapter (Chapter 11) covering both the math and descriptions.This is because most modeling of credit risks tends to be mathematically advanced. Itrelies on the already complex option-pricing theory. Like options, credit risks deal withcontingent delivery. However, the condition for payoff is not a tractable market price orrate movement, but rather a more esoteric concept of the change in the credit quality ofthe issuer (as evidenced by a ratings downgrade or default), which in turn depends onmostly unhedgeable variables (legal debt covenants, earnings performance, debt–equityratios, etc.).Each of the three parts of the book starts with a chapter containing a technical

primer, followed by more descriptive chapters containing applications of the analyticsin arbitrage-based trading strategies. The primers, labeled Financial Math I, II, and III,are intensely analytical, but at a mathematically low level. We avoid using calculus andinstead rely on numerical examples of real financial transactions. This should help notonly novice readers, but also readers with science backgrounds, who often follow theequations, but often find it difficult to relate them to actual money-making activities.The main quantitative tool used is cash flow discounting, supplemented by somerudimentary rate and price conventions. Chapter 2, ‘‘Financial Math I’’, contains abrief summary of present value techniques. It offers definitions of rates and yields, andintroduces the concept of a yield curve used to perform cash flow discounting. Itthen develops no-arbitrage equations for bond, stock, and currency markets. Chapter6, ‘‘Financial Math II’’, describes the mechanics of futures and forwards trading incommodities, interest rates, equities, and currencies. It then presents no-arbitrage rulesthat link forwards and futures back to spot markets. These rely on cost-of-carryarguments. Chapter 6 also develops further the concept of the yield curve by


showing how forwards are incorporated into it. Chapter 9, ‘‘Financial Math III’’, startswith basic payoff diagrams and static arbitrage relationships for options. It thendescribes the details of option valuation models that rely on the notion of dynamiccash-and-carry replication of option payoffs. It draws the fundamental distinctionbetween hedgers who manufacture payoffs and speculators who bet on future out-comes. It offers analytical insights into money-making philosophies of derivativestraders. The chapter covers stock, currency, and interest rate options.

The ‘‘Financial Math’’ primer chapters are followed by survey chapters that delvedeeper into the specifics of markets for different securities. These describe the players,the role of these markets in the savings–investment cycle, and the special conventionsand nomenclature used. They offer sketchy but insightful statistics. They also rely onthe knowledge contained in the primers to further develop detailed arbitrage strategiesthat are considered ‘‘benchmark’’ trades in each market. These are presented mostly aspure arbitrages; their relative value cousins are not difficult to imagine and some arealso described. In Chapter 3, we survey spot fixed income markets. We go over moneymarkets securities (under 1 year in maturity) which enjoy the most liquidity andturnover. We cover government and corporate bond markets. We touch on swaps,mortgage securities, and asset-backed securities. In Chapter 4, we describe themarkets for equities, currencies, and commodities. Stock markets are most likely themost familiar to all readers, so we focus on more recent developments in cross-listings,basket trading and stock exchange consolidations. Chapter 5 uses the mathematicalconcepts of Chapter 2 and applies them to the spot securities described in Chapters 3and 4. It presents pure arbitrage and relative value trades for different bond segments,equities, and currencies. It covers speculative basis trades in commodities. In Part II,following the primer on futures and forwards, Chapter 7 focuses on the cash-and-carryarbitrage and its various guises in currencies (covered interest rate parity), equities(stock index arbitrage) and bond futures (long bond futures basis arbitrage). It alsoextends the concept of hedging the yield curve using Eurocurrency strip trading andduration matching. Chapter 8 is devoted entirely to swap markets that represent spotand forward exchanges of streams of cash flows. These streams are shown to beidentical to those of bonds and stocks rendering swaps as mere repackagings of otherassets. This chapter combines the analytical treatment of swap mechanics with somemore descriptive material and market statistics. In Part III, following the optionsprimer, Chapter 10 describes a few forms of options arbitrage, admittedly in rathersimplistic terms, but it also extends the option discussion to multiple asset classes andoption-like insurance contracts. Part IV contains one chapter on credit risk and itsrelationship to fixed income assets described in prior chapters.

Clearly, the most important, but also the most quantitative chapters in the book areall the primers (Chapters 2, 6, 9, and 11). Readers interested only in the mechanics ofmarkets can read those in sequence and then use other chapters as reference material.For readers interested in the details of various markets, the order of study is veryimportant, particularly in Parts II and III. The descriptive chapters rely heavily onthe knowledge contained in their Financial Math primers. This is less so in Part I,where, apart from Chapter 5, we focus more on the description of basic securitiesand market infrastructure and less on arbitrage.

We hope our audience finds this book as contributing significantly to their deepunderstanding of today’s global financial marketplace.


____________________________________________________________________________________________________________________________________________ Part One ____________________________________________________________________________________________________________________________________________

_____________________________________________________________________________________________________________________________________________________________ Spot _____________________________________________________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________________________________________________________________ 2 __________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________ Financial Math I—Spot __________________________________________________________________________________

Cash flow discounting is the basis of all securities valuation. The fundamental value of astock is equal to the present value of all the dividends and capital gains the owner willbe entitled to in the future. The fundamental value of a bond is equal to the presentvalue of all the interest payments and principal repayment the owner will receive overthe life of the bond. These cash flows may be known in advance (principal of agovernment bond) or uncertain (capital gains on the stock). Market participants maydisagree in their estimates of the amounts. But the basic technique is always the same.Once you have determined the future cash flows, all you do is apply an appropriateinterest rate to discount them to today. That rate reflects both the cost of money andthe degree of uncertainty about the exact amount of the flows. The sum of all thediscounted cash flows for a security (i.e., the present value) is what you should bewilling to pay for that security (i.e., the price).The main premise of discounting is the concept of the time value of money. Money

can be rented to a bank to earn interest. Money can be rented from a bank by agreeingto pay interest. You would always pay less than $100 today for a promise of $100 in thefuture. This is not only because of a risk of the promise, but also because it would costyou less than $100 today to buy an investment that, with interest earned, wouldproduce $100 in the future. Whether you use owned funds or borrowed funds topurchase an investment is not important. If you use borrowed funds you pay explicitinterest. If you spend owned funds, you forgo the interest that you could have earned.This forgone interest is commonly referred to as an opportunity cost and it is as real acost as the interest paid on borrowed funds.College textbooks apply the notion of time value of money to single and multiple

cash flow securities, like stocks and bonds, as well as capital-budgeting projects. We willdevote a page to the review of that material; the reader is encouraged to study it more ifnecessary. Implicitly, many textbooks restrict themselves to using only one type ofinterest rate: the discount rate or a zero-coupon rate. That is, to compute the presentvalue of each cash flow, they assume that interest compounds and accrues annually overthe life of the cash flow and is due once at the end. This is just the tip of the iceberg.Interest can accrue and be paid according to many conventions. It may compound atone frequency (quarterly), but be paid at another (annually). The calendar for intra-year calculations may assume actual or even numbers of days per month or year. Therate of interest may be different for different future dates. Cash flows may not be due ateven intervals. The complications abound.To make sense of them, we review common compounding and day-count conven-

tions and then we look at three main arrangements, dating back to Phoenician times, ofhow interest can be earned and paid:

. Zero-coupon (discount or add-on), with no intervening interest cash flows.

. Coupon, with periodic fixed or variable interest payments.

. Amortizing, with periodic interest payments and partial principal repayments.

Typically, textbooks dealing with capital budgeting cover the first arrangement, fixed-income books the second, and mortgage books the third, reflecting where the differentforms of interest are most common. The rates quoted on these three different bases arenot directly comparable with each other, even after conversion to the same day-countand periodicity. However, the three are mathematically related and can be computedfrom each other through arbitrage arguments.

2.1 INTEREST RATE BASICS

We start with present values (PVs), compounding rules, and day-count conventions.

Present value

Suppose you earn interest on $500 at 5%. How much will you have in 1 year? Theanswer is $500 plus 5% of $500, or 500ð1þ 0:05Þ ¼ $525. How much will you have ifyou invest $500 for 2 years? In year 1, your investment will accrue to $525, but, in year2, you will earn interest on interest. You will get $525 plus 5% of $525, where each $525is equal to 500ð1:05Þ. That is, after 2 years, you will have 500ð1:05Þ þ 0:05 � 500ð1:05Þ ¼500ð1:05Þ2 ¼ $551:25. Generally, if you invest PV0 today at interest rate r for n years,your investment will have a future value of:

FV ¼ PV0ð1þ rÞn

Let us reverse the question. How much would you have to invest today so that at aninterest rate of 5% it would accrue to $500 2 years from today? Now, $500 is the futurevalue and we need to solve for PV0 in the above equation to get:

PV0 ¼ 500

ð1þ 0:05Þ2 ¼ $453:51

If you had easy access to borrowing and lending at 5%, you would be indifferentbetween $453.51 today and $500 2 years from today. You could convert one into theother at a known conversion ratio. In general, we can write an expression for the presentvalue (PV) of a single future cash flow as:

PV0 ¼ FV � 1

ð1þ rÞn

The expressions ð1þ rÞn and 1

ð1þ rÞn are referred to as future value interest factors and

present value interest factors, respectively, for a rate r and a number of periods n.Graphically, we can present our situation as:

0Today

1 2 3 n� 1 n

PV0 FV


Now suppose we ask a slightly more difficult question. How much would you have toinvest today in an account paying 5% interest, so that you could withdraw from theaccount $500 every year for the next 4 years? The first withdrawal is 1 year from todayand there is no money left in the account after the last withdrawal 4 years from today.Graphically, we can present this situation as:

To solve, we can imagine that, if we had money today, we could divide it into fourseparate investments each generating a $500 cash flow at different times. That is, wecould divide it into these amounts:

PV0 ¼ 500 � 1

1þ rþ 500 � 1

ð1þ rÞ2 þ 500 � 1

ð1þ rÞ3 þ 500 � 1

ð1þ rÞ4The total amount to be invested today is:

PV0 ¼ 500

�1

1þ rþ 1

ð1þ rÞ2 þ1

ð1þ rÞ3 þ1

ð1þ rÞ4�¼ $1,772:98

A constant cash flow (CF) over n periods starting one period from today is referred toas an ordinary annuity. Its present value is:

PV0 ¼ CF

�1

1þ rþ 1

ð1þ rÞ2 þ � � � þ 1

ð1þ rÞn�

and the expression in brackets is referred to as a present value annuity factor at a rate rfor a number of periods (years) n.The present value of a stream of cash flows can be interpreted as today’s value of the

promise of the entire future stream. It is an amount that would make one indifferentbetween (1) receiving it in one lump sum of PV0 today and (2) in the form of a cash flowstream in the future.

Compounding

When interest compounds intra-year instead of year to year, the calculations become alittle more complicated.

Annual example

On June 1, we invest c¼ 1,000 in a 1-year certificate of deposit (CD) yielding 3.25%. OnJune 1 of next year, we will get 1,000ð1þ 0:0325Þ ¼ c¼ 1,032.500.

Quarterly example

Suppose instead, on June 1, we invest c¼ 1,000 in a 3-month CD yielding 3.25%. Howmuch will we get on the due date of September 1?The first thing we need to keep in mind is that the rate of interest is always stated per

annum. The rate of 3.25% (p.a.) as quoted, first has to be de-compounded to obtain the

0 1 2 3

PV0 ¼ ?

4

500500 500 500

Financial Math I—Spot 29

interest rate per quarter. This can be done by dividing the rate by the number of interestperiods per year. Once we have done that, the answer becomes simply1,000ð1þ 0:0325=4Þ ¼ c¼ 1,008.125.

Quarterly rollover example

Suppose we invest c¼ 1,000 for a total of 1 year by first investing it in a 3-month CD. Weassume that CD rates do not change over the next year and we roll over the principalplus interest every 3 months into new CDs yielding 3.25% (p.a.). Our reinvestmentdates are: September 1, December 1, and March 1. The final maturity is June 1. Howmuch will we get a year from today? The answer is a compound formula of1,000ð1þ 0:0325=4Þ4 ¼ c¼ 1,032.898. We will get c¼ 1,000 principal and c¼ 32.898 in totalinterest.

Equivalent annual rate

We define the equivalent annual rate (EAR), in the CD example equal to 3.2898%, asthe rate that would have had to be offered on an annual investment to generate thesame amount of interest over 1 year as the compound investment at a quoted rate. If wedenote the quoted rate by r in percent per annum, and the number of compoundingperiods per year as m, then the relationship between the quoted rate r and the EAR is:

ð1þ r=mÞm ¼ 1þ EAR

For example, for a semi-annual rate m ¼ 2, while for a monthly compounded ratem ¼ 12. When comparing yields on investments of different compounding frequency,we convert the stated rates to EARs. As we will see, even that is not enough.

Day-count conventions

Each fixed-income market has its own quote and day-count convention. Our exampleswere simplified so that for quarterly periods we divided the stated rates by 4. Implicitly,we were using what is called a 30/360 day-count convention. A day-count convention is acommonly accepted method of counting two things: the number of days within theinterest calculation period and the number of days in a year. Under the 30/360 con-vention all months are assumed to have exactly 30 days and each year has 360 days, sothat each 3-month period represents exactly one-quarter of a year. In the above calcula-tions, the division by 4 and raising to the power of 4 were shortcuts for multiplying thequoted rates by 90/360 (i.e., computing the actual interest rate per 90-day period) andraising the gross de-compounded return (the one-plus-interest expression) to the powerof 360/90 (i.e., compounding it as many times as there are periods). In the rolloverexample, we could have written 1,000ð1þ 0:0325 � 90=360Þ360=90 ¼ c¼ 1,032.898. We alsoshould have amended the EAR definition to:

�1þ r � dayscalc

daysyear

�daysyear=dayscalc

¼ 1þ EAR


where dayscalc is the number of days in the interest calculation period and daysyear is theassumed number of days in a year.1

Many deposits, notably LIBOR (London interbank offered rate)-based Eurodollars,use an Act/360 day-count convention. For our 3-month (June 1–September 1) CD, theAct/360 convention means that the numerator is the actual number of calendar days inthe interest period, and in the denominator we assume that a year has exactly 360 days,and not 365 or 366 as it may be. On September 1, our 3-month CD would return theprincipal and interest worth 1,000ð1þ 0:0325 � 92=360Þ ¼ c¼ 1,008.306. The 1-year(June–June) rollover strategy would pay:

1; 000

�1þ 0:0325 � 92

360

��1þ 0:0325 � 91

360

��1þ 0:0325 � 90

360

��1þ 0:0325 � 92

360

�

¼ C=1; 033:361

as there are 92, 91, 90, and 92 days, respectively, in each 3-month reinvestment period.The EAR is equal to 3.3361%. LIBOR-based sterling deposits use an Act/365 basis,where we compute the actual number of days in the interest calculation period, butalways assume 365 days per year even if it happens to be 366.Let us consider one last ‘‘wrinkle’’. Suppose we deposit money on June 1 for the

period ending December 15. How much will we get back on the due date if we areearning 3.38% quarterly compounded on an Act/360 basis? The following expressiondescribes our accrual on a c¼ 1,000 investment:

1; 000

�1þ 0:0338 � 92

360

��1þ 0:0338 � 91

360

��1þ 0:0338 � 14

360

�¼ C=1; 018:593

Most commonly used day-count conventions include Act/365, Act/Act, Act/360 and 30/360. Government bond markets typically follow one of the first two, money marketsfollow the third, and corporate bond markets use the last one. There are exceptions tothese rules, and OTC derivatives markets can follow ‘‘wrong’’ conventions for differentunderlying securities. Many unsuspecting investors have been burnt in the past byruthless dealers playing day-count tricks!In this book, except when explicitly noted, we will assume the simple 30/360

convention. We will end up with more familiar divisions by 4, 2 or 1 for quarterly,semi-annual, and annual periods.

Rates vs. yields

Throughout the text, we use the word rate to denote the interest percentage that istypically explicitly stated and is used in a convention to compute the actual monetaryamount of interest paid to the holder of a security. We will use the word yield to denotethe interest percentage earned on the amount invested. The market sometimes uses thetwo interchangeably, but they should not be. A bond with a 5% coupon rate selling atpar (price equal to 100% of the face value) yields 5%. A bond with a 5% coupon rateselling below par (price less than 100% of face) yields more than 5% as the buyer has tospend less than the principal value of the bond.


1 Actually, even that formula is not general enough. The best reference on the subject is Marcia Stigum and Franklin L.Robinson, Money Market and Bond Calculations, 1996, Irwin, Chicago.

The definition of EAR has the word rate in it, even though it is more akin to a yield,because EAR is typically computed upfront, using a stated rate and assuming apurchase at par. In the above examples, if the 1-year c¼ 1,000 CD with a quarterlyrate of 3.25% were purchased for c¼ 990, then the quarterly equivalent yield (QEY)could be defined implicitly by the equality 990ð1þQEY=4Þ ¼ c¼ 1,032.50.

The investor would be said to earn a yield of QEY ¼ 17:7% on a c¼ 990 investment. Abond-equivalent yield (BEY) is the yield earned on buying an investment at its marketprice and collecting the cash flows as defined by the stated rate, but restated in afrequency convention of the bond market (e.g., semi-annual in the U.S. and theU.K. but annual in some continental European markets). That is, it is an annualequivalent yield (AEY) or semi-annual equivalent yield (SAEY), grossed up or downfrom a yield expressed in a natural frequency of the investment. The quarterly yield onthe CD in our example could be grossed up to arrive at a semi-annual bond-equivalentyield by pretending that the CD is rolled over for another quarterly period; that is:

ð1þQEY=4Þ2 ¼ ð1þ SAEY=2Þ

2.2 ZERO, COUPON, AND AMORTIZING RATES

Next we review the distinction between zero, coupon, and amortizing rates. Zeros arethe purest form of discounting rates in the sense that they translate directly any futurecash flow into its present value. A stream of cash flows is discounted by applying anappropriate zero rate to each cash flow to compute its present value and then bysumming the individual present values to obtain the present value of the entirestream. This can be done using one of the blended rates: coupon or amortizing. Inthat case, however, while the present value of the stream may be correct, the presentvalue of each individual cash flow is not, as it has the wrong rate applied to it. Theblended rate is imaginary; there is no investment accruing interest at that rate togenerate any one of the cash flows in the stream. Let us go into more detail.

Zero-coupon rates

All the examples so far involved zero-coupon rates2 (commonly referred to as discountrates as they can be used directly in cash flow discounting). These are earned oninvestments for which the accrued interest is received only once with the principalrepayment on the maturity date. The investment does not generate any intermediatecoupon interest cash flows (i.e., it has a zero coupon). The 3-month CD investment wasa zero-coupon investment over one quarter. The 1-year rollover strategy consisted offour consecutive quarterly zero-coupon investments. The investor received no cashflows during the entire year. All interest and principal were fully reinvested everyquarter. Graphically, the simplest way to present a zero-coupon rate is as follows:

00

1m

22m

33m

n� 1 nnm

PV0 CF� ð1þ r=mÞnm ¼Years

Sub-annual periods, if any


2 Zero-coupon rates are also referred to as spot rates. This is very misleading, since all rates (zero, coupon, and amortizing)can be spot (i.e., for an interest period starting now and ending in the future). They all can also be forward (i.e., for an interestperiod starting in the future and ending in the future).

Notice the absence of any intervening cash flows between today and the future datet ¼ nm. The 1-year rollover strategy has zero net intervening cash flows as both theprincipal and interest are fully reinvested each quarter. This is shown in the followingpicture (þ denotes an inflow, � denotes an outflow):

Zero-coupon interest can accrue on either an add-on or a discount basis. The distinc-tion here is only of the form, not of substance. In the add-on case, the investment ispurchased for a full face value, which is a multiple of some round number, and interestaccrues based on the principal equal to the face value of the security. In our retail CD,the investor deposited c¼ 1,000 and received, 3 months later, the principal and interestworth of c¼ 1,008.306. In contrast, most short-term securities sell at a discount from facevalue, and the interest rate is only implied by the ratio of the round-numberedface value and the purchase price of the security. For example, the price of 99.05for a 3-month U.S. Treasury Bill is expressed as percent of par. For a T-Bill promisingto repay exactly $10,000 in 3 months, the investor pays $9,905. The implied yieldon a 30/360 basis he earns can be obtained by solving the expression9905ð1þ r � 90=360Þ ¼ 9905ð1þ r=4Þ ¼ 10; 000 for r ¼ 3:8364%. Similarly, a buyer ofa 6-month U.S. T-Bill paying 98.1179 would compute his implied yield by solving in theexpression 98:1179ð1þ r=2Þ ¼ 100 for r ¼ 3:8364%. Even though the two T-Bills havethe same 3.8364% yield, they are not truly comparable: one matures in 3 months, theother in 6. Depending on at what rate we can reinvest the 3-month T-Bill, we could endup with more or less than the principal and interest on the six-month T-Bill in 6 months.Assuming no change in rates, we can use EAR as a comparison tool. To obtain EARs,we compute the yields each investment earns if it is rolled over at its original yield for atotal holding period of 1 year.

EAR3m ¼ ð1þ 0:038364=4Þ4 � 1 ¼ ð100=99:05Þ4 � 1 ¼ 3:8920%

EAR6m ¼ ð1þ 0:038364=2Þ2 � 1 ¼ ð100=98:1179Þ2 � 1 ¼ 3:8732%

Alternatively, we could compute the semi-annual equivalent of the 3-month rate byfollowing the logic behind EAR but only over a 6-month horizon. The equivalentsemi-annual rate (ESR) of the stated 3-month rate, r3m ¼ 3:8364%, would satisfy theequation ð1þ r3m=4Þ2 ¼ 1þ ESR=2. This is the essence of the BEY whose precisedefinition varies by market. Let us re-emphasize that, upfront, the realized yield onany rollover strategy is not known, as the reinvestment rate can change. The equivalentyield calculation relies on the unrealistic assumption that the reinvestment rate isknown and will not change.

Coupon rates

The word coupon comes from a physical piece of paper bond investors used to clip offthe bond to send to the bond issuer (borrower) to claim their periodic interest receipt.Today, only some Eurobonds come in bearer form with physical coupons. All U.S.

�P0

0

þP3m þ I3m�P3m � I3m

3 months

þP6m þ I6m�P9m � I9m

þP9m þ I9m�P9m � I9m þP12m þ I12m

6 months 9 months 12 months


government and corporate bonds and all European government bonds are registeredand often exist only ‘‘virtually’’ as computer entries of the owners’ names andaddresses. But the way interest is paid on bonds is the same as ever. Once or twice ayear, depending on the coupon frequency stated on the bond, the owners receive a cashflow equal to the coupon rate times the face value of the bond they hold multiplied bythe appropriate day-count fraction.

Let us consider an example. On June 30, 2004 an investor purchases a 10-year bondwith a face value of £2,500 and a stated annual coupon rate of 4.5%. The maturity dateis June 30, 2014 and coupons accrue from June 30 to June 30. On June 30 of each yearbetween 2005 and 2014, the bond holder receives interest payments equal to2,500 � 0:045 ¼ £112.50. Additionally, on June 30, 2014 he receives his principal of£2,500 back. The actual CFs from the bond are portrayed in the following picture:

Most commonly, CFs are represented as percentages of par, as in the followingnormalized picture:

The main difference between a coupon bond and a zero-coupon bond is the stream of4.5% or £112.50 cash flows in years 2005–2014. These are not simply accrued and rolledup until the final maturity date, but physically paid out. A zero holder would onlyreceive one large payment at maturity, on June 30, 2014, which would consist of theface value repayment of £2,500 and a 10-year accumulation of interest.

Coupon yields and rates can be expressed on a variety of compounding and day-count bases, typically following a particular convention. The bond debenture contract(fine print) always states clearly how the interest is accrued and paid. But the legallanguage can be far from plain English. The stated rate is always annualized and needsto be de-compounded. For example, a 6% quarterly bond does not pay a coupon equalto 6% of the face value every 3 months; rather, it pays a coupon close to but notnecessarily identical to 1.5% of the face value every 3 months. On a 30/360 day-count basis, it is exactly 1.5%. On any other basis, the numerator and the denominatorof the day count may not result in an exact division by 4 for all periods. The only thingreally ‘‘fixed’’ is the stated rate; the dollar interest payment may change each period. Atleast the language is consistent: the percentage rate is always expressed on a per annumbasis and must be adjusted by the compounding frequency to get the periodic cash flow.A 3-year 4.5% semi-annual coupon bond would have the coupon cash flows of4:5=2 ¼ 2:25% every 6 months and principal repayment flow of 100 in 36 months:

Notice that the 3-year 4.5% semi-annual coupon bond can be viewed as an ordinaryannuity of 2.25 over six periods and a one-time cash flow of 100 six periods from thepurchase date. When computing the present value of the bond, it is convenient to break

112.50 112.50 112.50 112.50 112.50 112.50 112.50 112.50 112.50 2,612.50

0 1 2 3 4 5 6 7 8 9 10

4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 104.5

0 1 2 3 4 5 6 7 8 9 10

2.25 2.25 2.25 2.25 2.25 (100þ 2.25)

0 6m 12m 18m 24m 30m 36m


it down into these two components. Suppose the interest rate with which to discount allthe cash flows is 5% semi-annual. To get the value of the bond, we would set up thefollowing equation:

PV0 ¼ 2:25�1þ 0:05

2

�þ 2:25�1þ 0:05

2

�2þ 2:25�

1þ 0:05

2

�3þ 2:25�

1þ 0:05

2

�4

þ 2:25�1þ 0:05

2

�5þ 102:25�

1þ 0:05

2

�6

which, collecting terms for the annuity component and the principal repaymentcomponent, looks like this:

PV0 ¼ 2:251�

1þ 0:05

2

�þ 1�1þ 0:05

2

�2 þ 1�1þ 0:05

2

�3 þ 1�1þ 0:05

2

�4 þ 1�1þ 0:05

2

�5

0BBB@

þ 1�1þ 0:05

2

�6

1CCCAþ 100�

1þ 0:05

2

�6

The value in this case is PV0 ¼ 98:6230 which is less than 100. We used a yield of 5.00%to discount the cash flows of a bond with a coupon rate of 4.5%. If the bond sells in themarket for 98.6230, then the 5.00% is the bond’s yield to maturity (see below).Zero-coupon bonds are special cases of coupon bonds with the coupon rate equal to

0. However, they should be viewed as fundamentally different from their couponcousins. Later in this chapter, we show how we can build a set of zero yields fromcoupon yields and vice versa, but before that let us attempt to define what a yield on acoupon instrument is.

Yield to maturity

A holding period return (HPR) is a theoretical single yield earned on purchasing aninvestment for a given price, receiving cash flows from it, if any, over some knownholding period and selling it at some known price at the end of that period. That is, theknowns are:

. Purchase price.

. Intermediate cash flows.

. Sale price.

In computing a yield over a holding horizon, one could make an assumption that thecash flows obtained from the investment have been re-invested at different rates. ForHPRs, the reinvestment rate is assumed to be constant and equal to the HPR, but theholding period may be different from the maturity of the instrument (e.g. a 2-year


holding period for a 5-year bond or a 3-year holding period for a stock with infinitematurity). HPRs can be expressed on any compounding and day-count basis.

A yield to maturity (YTM) is a holding period return over a holding period equal tothe maturity of the instrument. All the cash flows are assumed to have been receivedand reinvested at the rate equal to the holding period return. The assumption that theinstrument is held to maturity also ensures that the sale price is equal to the face value.YTMs can be expressed on any compounding and day-count basis. Most dealersconvert non-native YTMs to a bond-equivalent basis which, in the U.S., is defined asthe SAEY on an Act/Act (governments) or 30/360 (corporates) basis, and, in most ofEurope, as the AEY on an Act/Act basis. We illustrate the concepts of a holding periodreturn and a YTM through some examples.3

Suppose 2 years ago you purchased a 2-year discount (zero-coupon) bond for 91.00.Today it matures and you receive 100.00 back. What semi-annual YTM have youearned? We solve for the implied zero rate r such that 91:00ð1þ r=2Þ4 ¼ 100. We getr ¼ 4:7716%. The YTM on a zero-coupon bond is equal to the zero rate itself as thezero, by definition, has no reinvestment. Now let us look at the coupon bond.

Suppose 2 years ago you purchased a 2-year 4% semi-annual coupon bond for99.0538. Today it matures and you receive 100.00 back. What semi-annual YTMhave you earned? Let us examine the cash flows promised by the bond. We had anoutflow of $99.0538 upfront, four inflows of 0:04 � 100 � 1

2¼ $2:00 on each coupon date

and an inflow of $100 on the final date (today).Textbooks offer the following definition of YTM: YTM is equal to an interest rate

such that if we discount all the cash flows from the bond at that rate, then the obtainedpresent value is equal to the price of the bond. This follows the logic of how we solve forthe YTM. The cash flows can be portrayed as:

To discount all the cash flows at some rate r to today, we would set up the followingequation:

PV0 ¼ 2:00�1þ r

2

�þ 2:00�1þ r

2

�2 þ 2:00�1þ r

2

�3 þ 102:00�1þ r

2

�4

Then we would solve for r such that the present value is equal to the initial investmentPV0 ¼ 99:0538. We could use a financial calculator (solving for the equivalent notion ofthe internal rate of return, or IRR), use a polynomial root finder, or solve by trial anderror in a spreadsheet. (For example, we would try 4% and 5% as two possiblesolutions: the first would result in a PV > 99, the second in a PV < 99. We couldthen try a rate between 4% and 5% to get closer to the true value of 99.0538. And

0

2.00 2.00 2.00 (100þ 2.00)

6m 12m 18m 24m


3 All of our examples assume that the investment is purchased a moment after a coupon has been paid. That is, we do notneed to make here, and in fact throughout the entire book, a distinction between the dirty price and the clean price. When acoupon bond is purchased between coupon payment dates, the buyer pays the seller the so-called dirty price, which includesan allowance for the interest that has accrued between the last coupon date and the purchase date. The seller is entitled to thatinterest, but by surrendering the bond, he has no possibility to collect it. The buyer will receive it included in the wholecoupon payment on the next coupon date. The so-called clean price is equal to the dirty price minus the accrued interest.

so on.) We can verify that the solution is r ¼ 4:5% by substituting 4.5 for the rate andsolving to get 99.0538 as the PV.The calculation mechanics hide the true nature of YTM as a ‘‘blended’’, or

‘‘average’’, yield actually earned on the investment. Let us imagine that we have amoney-market account that pays a semi-annually compounded rate r ¼ 4:5% on anydeposits into it. Suppose 2 years ago we deposited into that account $99.0538. On eachcoupon date, we withdrew $2.00 from the account. At maturity, we withdrew $100. Wecan show that if the interest rate the money-market account paid on any balances left init was equal to the computed YTM of 4.5%, then we could make the withdrawals andend up with no balance in the account at the end.We deposit $99.0538 into the account. At 4.5% semi-annually, 6 months later we

have at our disposal 99:0538

�1þ 0:045

2

�¼ 101:2825. Of that, we pay ourselves $2.00.

We redeposit the remaining $99.2825 for another 6 months at 4.5% semi-annually to

get 99:2825

�1þ 0:045

2

�¼ 101:5164 in 12 months. We pay ourselves $2.00 and rede-

posit the remaining $99.5164 for another 6 months at 4.5% semi-annually to get

99:5164

�1þ 0:045

2

�¼ 101:7555 in 18 months. We pay ourselves $2.00 and redeposit

the remaining $99.7555 for another 6 months at 4.5% semi-annually to get

99:7555

�1þ 0:045

2

�¼ 102:00 in 24 months. We withdraw $102.00 and close the

account with a balance of $0.00. We repeated the wording of ‘‘reinvesting foranother 6 months at 4.5% semi-annually’’ to stress the following point:

Conclusion 1 YTM is the assumed constant reinvestment rate over the life of thebond.

There is another interpretation of YTM. Suppose we divide the original sum of$99.0538 into four separate investments:

. A 6-month CD of $1.9560 yielding 4.5% semi-annually compounded.


. An 18-month CD of $1.8709 yielding 4.5% semi-annually compounded.


The reader should convince himself that the CD investments will accrue to $2.00, $2.00,$2.00, and $102 at their respective maturity dates (i.e., their cash flows will matchexactly those of the coupon bond). This leads us to the following point:

Conclusion 2 YTM is the single zero-coupon investment rate for the maturity datescorresponding to all the cash flows over the life of the bond.

That is if all the term zero rates to different dates are assumed equal! In reality, both thereinvestment rates and the term rates are not likely to be constant or equal to eachother. If we were to divide our $99.0538 into four different amounts in order to replicatethe cash flows of the coupon bond, these amounts would have to be invested using theactual zero-coupon rates (term lending rates) in the market. The investment amountswould all be different from the ones arrived at using the YTM for discounting. YTM isa fictitious rate. It is a mathematical construct. It is the average rate earned over the life


of the bond held to maturity given the price paid. That is why we called it ‘‘blended’’ or‘‘average’’.

Amortizing rates

Another way of earning interest, in addition to zero and coupon rates, is throughamortizing interest rates. Some bonds and most mortgage loans follow this arrange-ment. Like the fixed coupon bond issuer, the amortizing rate borrower agrees toconstant periodic cash flows over the life of the loan. But each cash flow consists ofan interest portion and a principal repayment portion so that the borrower does nothave to repay the entire principal at the maturity of the loan. Instead, he repays it pieceby piece with each periodic payment. To distinguish the amortizing loan from a couponloan, the latter is often referred to as a balloon loan and the principal repayment atmaturity as the balloon payment.

The best example of an amortizing rate loan is a 30-year fixed-rate mortgage. Itconsists of 360 equal monthly payments (30 years� 12 months each). The mortgageborrower obtains the full amount of the loan upfront with which he pays for a piece ofreal estate. The monthly payments cover both the interest on the loan and the repay-ment of the principal. Financial calculators and computer spreadsheets offer built-infunctions to compute the constant monthly payment. They can also help construct theso-called amortization table which breaks down each payment into its interest andprincipal components. What is immediately clear from such a table is that over timethe interest portion of the level payment decreases while the principal portion increasesas the loan is paid down. This is obvious as each month interest is paid on thedecreasing outstanding principal (i.e., the yet-to-be-repaid part), not the originalamount of the loan. Interest and principal portions balance each other in such a waythat the total payment remains constant.

An amortization schedule for a mortgage looks quite complicated. We can easily losesight of the simple nature of the amortizing loan. It is nothing but an ordinary textbookannuity. Let us illustrate this point on an example. Consider a 2-year, 4.5%, semi-annual interest amortizing loan of $100. The repayment consists of four equal paymentsin 6, 12, 18, and 24 months such that the PV of those payments is equal to the face valueof the loan. These can be portrayed as:

To solve for the constant payment of cash flow (CF) we have to set up the followingequation:

100 ¼ CF�1þ 0:045

2

�þ CF�1þ 0:045

2

�2þ CF�

1þ 0:045

2

�3þ CF�

1þ 0:045

2

�4

and we can group the terms:

100 ¼ CF1�

1þ 0:045

2

�þ 1�1þ 0:045

2

�2þ 1�

1þ 0:045

2

�3þ 1�

1þ 0:045

2

�4

0BB@

1CCA

0 6m 12m 18m 24m

CFCF CF CF


to see that the expression in parentheses is a 4-period annuity factor with an interestrate 2.25% per period. The solution is: CF ¼ 26:4219.Let us also illustrate the logic of interest and principal component calculation. For

the first 6 months the interest portion is0:045

2� 100:00 ¼ 2:25. The remainder

26:4219� 2:2500 ¼ 24:1719 goes toward the principal reduction. Right after the firstpayment, the outstanding principal is equal to 100:0000� 24:1719 ¼ 75:8281.Interest for the period starting in 6 months and ending in 12 months is equal to

0:045

2� 75:8281 ¼ 1:7061. The outstanding principal gets reduced by 26:4219� 1:7061 ¼

24:7158 to 75:8281� 24:7158 ¼ 51:1123.

Interest for the period starting in 12 months and ending in 18 months is equal to0:045

2� 51:1123 ¼ 1:1500. The outstanding principal gets reduced by 26:4219� 1:1500 ¼

25:2719 to 51:1123� 25:2719 ¼ 25:8404.

Interest for the period starting in 18 months and ending in 24 months is equal to0:045

2� 25:8404 ¼ 0:5814. The outstanding principal is reduced by 26:4219� 0:5814 ¼

25:8405 to 25:8404� 25:8405 ¼ 0:0001 � 0:0000. The loan balances are summarized inTable 2.1.

Table 2.1 Amortization table, $100 loan, maturity 2 years, 4.5% semi-annual rate

Period Start principal Payment Interest portion Principal portion End principle

1 100.0000 26.4219 2.2500 24.1719 75.82812 75.8281 26.4219 1.7061 24.7158 51.11233 51.1123 26.4219 1.1500 25.2719 25.84054 25.8405 26.4219 0.5814 25.8405 0.0000

Floating-rate bonds

So far, we have considered only bonds with interest rates that are known in advance.Zero, coupon, or amortizing rates uniquely determine the cash flows a bond will payover its life to maturity. Many bonds, however, pay cash flows that are not known inadvance, but are instead tied to some interest rate index. The cash flows of these bondscan be present-valued only by use of rollover arguments. As a way of introduction tovaluation of securities with unknown cash flows, we consider a floating rate bond.Suppose ABC Inc. issues a 5-year $100 annual coupon bond. The coupon for each

annual interest period will be set equal to the 1-year interest rate in effect at thebeginning of that interest period and will be paid at the end of the interest period.The first coupon to be paid in 1 year is set today, at the time of the issue. The secondcoupon rate will be set 1 year from today and paid 2 years from today. It will be setequal to the then prevailing 1-year interest rate. The third coupon will be set 2 yearsfrom today and paid 3 years from today, and so on. How much will investors pay forsuch a bond?


In order to answer that question, we make a few observations. The coupon for eachinterest period will be ‘‘fair’’. It will change every year and it will reflect the cost of‘‘renting’’ money for that year. The timing of the coupon-setting process corresponds toa sequence of new 1-year loans. On such loans, the rate would be set at the beginning ofthe year and paid at the end (i.e., it would be known in advance). The floating rate bondis equivalent to a revolving loan. Each year, ABC can be viewed as paying off a 1-yearbond issued a year earlier and refinancing with a new issue of a 1-year bond.

We can use these observations to recursively deduce the price of the floating ratebond. Consider owning the 5-year bond 4 years from today, just after a couponpayment. The bond has 1 year left and the rate set is equal to ~xx4�5. The presentvalue of your remaining cash flow 1 year hence is:

P4 ¼ 100þ ~xx4�5 � 1001þ ~xx4�5

¼ 100

That is, on an investment of $100, you will be paid a rate ~xx4�5, which is also the 1-yeardiscount rate at that time. Knowing that the bond will be worth 100 in year 4, the valueof the bond in year 3 will be equal to the discounted value of the coupon of ~xx3�4 � 100set then, to be paid 1 year hence, plus the $100 you will be able to sell the bond for 1year hence. That is, the price 3 years from today will be:

P3 ¼ ~xx3�4 � 100þ P4

1þ ~xx3�4¼ ~xx3�4 � 100þ 100

1þ ~xx3�4¼ 100

The recursive argument continues all the way to year 1 for which the interest rate isknown and equal to the 1-year discount rate.

Today, the price of the floating rate bond is equal to 100. That price will stay close topar throughout the life of the bond deviating only slightly inside the interest periodsand returning to par right after each coupon payment.4

2.3 THE TERM STRUCTURE OF INTEREST RATES

At any given moment, zero rates, coupon rates, and amortizing rates are not equal toeach other. Within the same category, interest rates for different maturities are notequal to each other. Not only are the different rates not equal to each other, butthey also change all the time!

A graph of market interest yields against their maturities is called the term structureof interest rates, or the yield curve. The graph is a snapshot of the market YTMs for agiven moment in time. What is plotted on the y-axis is yields and not rates as the firstdefinition may misleadingly imply. The x-axis contains maturities relative to the date ofthe graph (e.g., 6 months from today, 1 year from today, 5 years from today, etc.). Theyields can be on any day-count/compounding basis, but this is most commonly dictatedby convention.

As there are many types of interest rates, there are many term structures. Bonds fordifferent issuers which represent the different credit quality of the issuers can begrouped into rating categories. In that sense, we can talk about the BBþ term structure


4 This assumes that the credit quality of the issuer remains the same throughout the life of the bond.

or the A� term structure. Even for each issuer, we can have many term structures aseach bond may be collateralized differently. For example, U.S. municipalities issueeither general obligation bonds, guaranteed by the general credit of the municipality,or revenue bonds, guaranteed by the revenues of a particular project like highway tolls.The two categories may enjoy different credit spreads and overall interest rates even forthe same maturity. Abstracting from all these credit issues, generally, we look at twotypes of term structures:

. The term structure of discount rates (i.e., zeros).

. The term structure of par rates (i.e., yields on coupon bonds that trade close to par).

In the U.S., the most commonly watched term structure is that of on-the-run T-Bills.These are newly issued T-Bills, notes, and bonds with a few standard maturities.Because they are newly issued, they trade very close to par as the coupon rate is setclose to the market yield, and they are very liquid as there is great interest in themamong investors. On September 18, 2002 the Treasury yields in Table 2.2 werereported.

Table 2.2 U.S. Treasury bonds

Maturity Yield Yesterday Last week Last month

3 Month 1.54 1.55 1.56 1.526 Month 1.56 1.57 1.60 1.572 Year 1.95 1.98 2.15 2.205 Year 2.88 2.88 3.12 3.3710 Year 3.81 3.81 4.05 4.2830 Year 4.71 4.72 4.87 5.04

Source: http://bonds.yahoo.com/rates.html on September 18, 2002.

From this information we can produce the term structures for the four dates on onegraph as in Figure 2.1.

0

1

2

3

4

5

6

0 10 20 30

Term

Yield

18/09/02

17/09/02

11/09/02

18/09/01

Figure 2.1 The term structure of treasury rates on September 18, 2002.

The graph illustrates how the par Treasury rates for different maturities had risen priorto September 18, 2002, but perhaps not equally; the greatest absolute increase being inthe long end of the curve.


On any given date the slope of the curve can be smooth, jagged, humped, etc., as itreflects real market variables. A few shapes are of interest: upward sloping, downwardsloping (inverted), or flat. These do not have precise definitions except as implied by thenames. Market analysts and economists often aver that the shape of the yield curvepredicts the economic cycle to follow. For example, upward sloping is purported tosignal expansion; inverted, or downward, sloping a recession; and inverted with a highshort rate perhaps a currency crisis. We show some examples in Figure 2.2.

0

1

2

3

4

5

6

0 10 20 30

Term

Yield

18/09/02

17/09/02

11/09/02

18/09/01

0

1

2

3

4

5

6

7

8

0 10 20 30

Term

Yield

Upward

Downward

Flat

Inverted

Figure 2.2 Yield curve shapes.

The Treasury yield curve blends discount (i.e., non-coupon-paying) T-Bills withcoupon-paying Treasury notes and bonds. This is done for completeness as there areno 3-month coupon-paying Treasuries. For consistency, all rates are expressed as semi-annual bond equivalents.

Although most yield curves found in the press are those for coupon par instruments,the most useful one is the term structure of zero-coupon interest rates, commonlyreferred to as the discount curve, or the zero curve. Almost all trading desks computethe zero curve relevant for their market, because it allows them to simply read off therates that must be used for discounting cash flows scheduled for different dates. Whenfaced with valuing a new bond in the market, a quantitative analyst working for thedesk determines each cash flow of the bond and discounts it using the zero rate corre-sponding to the date of the cash flow. We explain in the next few sections why that is thecorrect procedure. We also explain how to construct the discount curve from observedmarket yields.

Discounting coupon cash flows with zero rates

Let us consider the following:

Problem We are considering buying c¼ 100 of a 4-year, 6%, annual coupon bondissued by a German company XYZ GmbH. The cash flows on the bond can be depictedas:

6:00þ 100:00 ¼ 106:00

4y

6.00

3y

6.00

2y

6.00

0 1y


Suppose we also observe that XYZ GmbH has four other zero-coupon bonds out-standing. Their maturities and yields are summarized in Table 2.3.

Table 2.3 XYZ GmbH’s maturities and yields

Maturity Zero yield

1 5.002 5.753 6.104 6.50

How much would we be willing to pay for the coupon bond?

Solution Suppose, instead of buying the coupon bond, we bought four zero couponsissued by XYZ GmbH, each with the face value and maturity matching the cash flowsof the coupon bond:

. For the 1-year zero with a face value of c¼ 6.00 we would pay6

ð1þ 0:05Þ ¼ 5:7143:


ð1þ 0:0575Þ2 ¼ 5:3653:


ð1þ 0:061Þ3 ¼ 5:0235:


ð1þ 0:065Þ4 ¼ 82:3963:

The total we would spend would be c¼ 98.50. Since the cash flows from the four invest-ments match exactly those of the coupon bond, we can argue that we would be willingto pay c¼ 98.50 for the coupon bond.

YTM relationship The YTM on the coupon bond happens to be ytm ¼ 6:4374%.That is, if we discount each cash flow at 6.4374% instead of their respective discountrates we would get the following answers:

. For the 1-year zero6

ð1þ 0:064374Þ ¼ 5:6371:


ð1þ 0:064374Þ2 ¼ 5:2962:


ð1þ 0:064374Þ3 ¼ 4:9759.


ð1þ 0:064374Þ4 ¼ 82:5901.

The sum equals c¼ 98.50. But the four ‘‘present values’’ obtained in this way are totallyfictitious. That is, they do not represent the amounts we would have to pay for realsecurities. We had obtained those by discounting at the zero rates of the four realdiscount bonds.


Once we know the discount rates for any maturity, we know how to discount cash flowsfor any coupon or amortizing bond. This is because we can replicate the cash flows ofthat bond with real zero-coupon securities. The discounting is not a mathematicalequation, but a reflection of a replicating strategy. Obtaining a zero curve greatlysimplifies the analysis of any new security.

In reality, many issuers tend to only issue coupon securities (i.e., zero rates are notobserved directly in the market). What can we do when this is the case?

Constructing the zero curve by bootstrapping

The discount curve can be obtained by a process known as a zero bootstrap. This takesas given the par coupon rates observed in the market and sequentially produces the zerorates one by one from the shortest to the longest maturity, just like lacing boots. Abootstrap looks messy on paper, but setting it up in a spreadsheet requires no skill. Theonly confusion usually has to do with day-count conventions. We will describe theprocess for annual rates.

Problem We observe the following annual coupon bonds that XYZ GmbH hasoutstanding in the market. These are newly issued bonds and some old bonds thatoriginally had much longer maturities.

Table 2.4 XYZ GmbH’s annual coupon bonds

Maturity Coupon Price

1 5.10 100.09522 5.60 99.76183 6.10 100.09624 6.00 98.4993

Can we deduce what the zero curve for XYZ GmbH is given this market information?

Solving for the 1-year zero rate Let us first look at the 1-year bond. There is only onecash flow from this bond equal to c¼ 105.10 at maturity. This can be depicted as:

Let us compare this with the 1-year c¼ 100 face value zero-coupon bond whose singlecash flow can be depicted as:

Investors would be indifferent between the 1-year 5.1% coupon bond with a face valueof c¼ 100 and a 1-year zero-coupon bond with a face value of c¼ 105.10. Both investmentswould have the same single cash flow of c¼ 105.10 in 1 year. Both are in fact purediscount bonds with a face value of c¼ 105.10. The coupon bond is just labeled

105.10

0 1y

100

1 1y


‘‘coupon’’ as that is presumably how it was originally issued. If the zero-coupon bondcame in denominations of c¼ 0.10, investors could simply ask for 1,051 such ‘‘bondlets’’to replicate the 1-year coupon bond. Given that the coupon bond and the zero-couponbond are perfect substitutes for each other, they must have the same yield. We caneasily compute the yield on the 1-year coupon bond implied in its market price bysolving the following equation for z1:

100:0952 ¼ 105:10

ð1þ z1ÞThe solution is z1 ¼ 5%. Any cash flow payable by XYZ GmbH in 1 year should bediscounted by 5%, because it could be replicated by holding the appropriate face valueof the 1-year coupon bond. In other words, any 1-year discount bond issued by XYZGmbH would have to yield 5%, because it could be replicated by holding the appro-priate face value of the 1-year coupon bond.The first step in the bootstrap is typically simple, because the shortest maturity

coupon bond may not have any intervening coupons left.

Solving for the 2-year zero rate Let us examine the 2-year coupon bond. It has acoupon rate of 5.60% and it sells for c¼ 99.7618. From that information, we candepict its cash flows as:

From the first step of the bootstrap, we know that we can compute the PV of the 1-yearcash flow by applying the discount rate z1 ¼ 5%. That is, we can get:

PVðCF1Þ ¼ 5:60

ð1þ 0:05Þ ¼ 5:3333

We also know that the total PV of the bond equal to its current price is c¼ 99.7618. Sothe present value of the second cash flow must be the difference:

PVðCF2Þ ¼ 99:7618� 5:3333 ¼ 94:4285

If the bond could be separated and sold as two separate strips, then the first one withthe face value of c¼ 5.60 would sell for c¼ 5.3333 and the second one with the face value ofc¼ 105.60 would sell for c¼ 94.4285. We can solve for the discount yield implied in theprice of the 2-year strip by solving the following equation for z2:

94:4285 ¼ 105:60

ð1þ z2Þ2to get z2 ¼ 5:75%. If XYX GmbH had a 2-year zero outstanding, then it would have toyield 5.75%; otherwise, investors could replicate its cash flow of c¼ 100 in 2 years bybuying the appropriate face value of the second strip.

The arbitrage argument But suppose that the 2-year coupon bond cannot be separatedand sold as two strips. Can we still claim that the 2-year discount rate should be 5.75%?This is actually the crux of the argument.

105.60

2y

5.60

0 1y


Suppose an investor wants to enter into a strategy in which a payment of c¼ 105.60 in2 years is promised to her by XYZ GmbH (i.e., she wants to replicate the non-existentsecond strip or synthetically create a 2-year zero). She could do it by entering into thefollowing two trades simultaneously:

. Buying c¼ 100 face value of the 2-year, 5.60% coupon bond for c¼ 99.7618.

. Shorting c¼ 5.60 face value of a 1-year zero for which she would receive c¼ 5.3333.

Her net investment is c¼ 94.4285. Her cash flows can be portrayed as:

This matches exactly the 2-year strip. Thus, a 2-year zero investment can be synthetic-ally replicated by going long a 2-year coupon bond and short a 1-year zero-couponbond. Any 2-year zero-coupon bond issued by XYZ GmbH would have to yield at least5.75%; otherwise, nobody would buy it. All investors would simply enter into syntheticreplicating strategies.

Furthermore, if XYZ GmbH did issue a 2-year zero and it yielded more than 5.75%,all investors would buy that zero and simultaneously construct a short 2-year strip withoffsetting cash flows, thereby locking in riskless profits. The strategy could be summar-ized as the following three trades:

. Buy the 2-year zero with a face value of c¼ 105.60 for less than c¼ 94.4285 (i.e., yieldingmore than 5.75%).

. Short c¼ 100 face value of the 2-year, 5.60% coupon bond to receive c¼ 99.7618.

. Buy c¼ 5.60 face value of a 1-year zero by paying c¼ 5.3333.

In the short coupon bond transaction, the investor would borrow the bond from a thirdparty, sell it in the market for its current price, but would be obligated to replace thecoupons and principal repayment cash flows to the bond lender. The last two trades—short coupon bond and long 1-year zero—would bring a cash inflow of c¼ 94.4285 today.If the first one—long 2-year zero—costs less than c¼ 94.4285 to enter into, then thearbitrageur would have a positive cash flow today. Her future obligations would bematched at every point in time as shown below:

Long 2-year coupon

Short 1-year zero

Net

5.60

(5.60)

105.60

105.60

2y0 1y

Long 2-year zero

Short 2-year coupon

Long 1-year zero

Net

(5.60)

5.60

0

105.60

(105.60)

0

2y0 1y


All she would have to do in the future is to collect the receipts on her 1-year and 2-yearzero investments and use them to satisfy her coupon obligations to the lender of thebond. She would not be the only one pursuing this arbitrage strategy. All investorswould immediately pursue strategies like this one, driving the price of the newly issued2-year zero up and its rate down to the 5.75% level. It is also possible that the 2-yearcoupon rate might be driven up and the 1-year zero rate down at the same time. Allthree instruments would have to find a level at which arbitrage would be prevented andwe could safely apply our bootstrap math.One lesson from all of this is that the no-arbitrage principle requires that all instru-

ments be freely traded and both longs and shorts allowed. Our example also presumesthat the bid–ask spread (i.e., the difference in the prices at which the bonds are offeredto be bought and sold) is very narrow (we assumed a spread of zero). Illiquid marketswith wide bid–ask spreads also follow arbitrage rules, but the spread has to be explicitlytaken into account to compute arbitrage bands around mid-market prices and rates. Inliquid markets, the standard procedure is to use mid-market rates to compute theimplied mid-market zero rate and then adjust it to bid or ask.

Solving for the 3-year zero rate Let us examine the 3-year coupon bond. It has acoupon rate of 6.10% and it sells for c¼ 100.0962. From that information, we candepict its cash flows as:

We have solved for z1 ¼ 5% and z2 ¼ 5:75%, and we know how to discount the firsttwo cash flows of the bond:

PVðCF1Þ ¼ 6:10

ð1þ 0:05Þ ¼ 5:8095

PVðCF2Þ ¼ 6:10

ð1þ 0:0575Þ2 ¼ 5:4547

We also know that the total present value of the bond equal to its current price isc¼ 100.0962. So the present value of the third cash flow must be the difference:

PVðCF3Þ ¼ 100:0962� ð5:8095þ 5:4547Þ ¼ 88:8320

If the bond could be separated and sold as three separate strips, then:

. The 1-year strip with the face value of c¼ 6.10 would sell for c¼ 5.8095.



We can solve for the discount yield implied in the price of the 3-year strip by solving thefollowing equation for z3:

88:8320 ¼ 106:10

ð1þ z3Þ3

to get z3 ¼ 6:10%. If XYX GmbH had a 3-year zero outstanding, then it would have toyield 6.10%. It is a pure coincidence that the 3-year zero rate, or the 3-year zero yield, is

106.10

3y

6.10

2y

6.10

0 1y


equal to the 3-year coupon rate of the particular bond we used. The 3-year coupon yieldis not equal to 6.10% as the bond sells for a price higher than the par of 100. If XYXGmbH were to issue another 3-year coupon bond and wanted to sell it at par, then itscoupon would have to be set below 6.10%. We will come back to this point after thecompletion of the bootstrap.

The arbitrage argument Let us again show that the 3-year zero of 6.10% is not just aproduct of a mathematical procedure, but that market forces (i.e., real human behavior)will guarantee that level. We assume that the 1- and 2-year zero-coupon bonds issued byXYZ GmbH are actively traded in the market or they can be created synthetically bytaking simultaneous positions in existing coupon bonds and zero bonds.

Suppose an investor wants to enter into a strategy in which a payment of c¼ 106.10 ispromised to her by XYZ GmbH (i.e., she wants to synthetically create a 3-year zerowith a face value of c¼ 106.10). She could do it by entering into the following three tradessimultaneously:

. Buying c¼ 100 face value of the 3-year, 6.10% coupon bond for c¼ 100.0962.



Her net investment is c¼ 88.8320. Her cash flows can be portrayed as:

This matches exactly the 3-year strip. Any 3-year zero-coupon bond issued by XYZGmbH would have to yield at least 6.10%; otherwise, nobody would buy it. Allinvestors would simply enter into this synthetic replicating strategy with existing bonds.

Again, we can also argue the yield on a new 3-year zero could not be higher than6.10%. Suppose that XYZ GmbH did issue a 3-year zero yielding more than 6.10%. Allinvestors would buy that zero and at the same time construct a synthetic short 3-yearzero with offsetting cash flows to lock in riskless profits. The strategy could be summar-ized as the following four trades:

. Buy the 3-year zero with a face value of c¼ 106.10 for less than c¼ 88.8320 (i.e., yieldingmore than 6.10%).

. Short c¼ 100 face value of the 3-year, 6.10% coupon bond to receive c¼ 100.0962.



In the short coupon bond transaction, the investor would borrow the bond from a thirdparty, sell it in the market for its current price, but would be obligated to replace thecoupons and principal repayment cash flows to the bond lender. The last three trades—

Long 3-year coupon

Short 1-year zero

Short 2-year zero

Net

6:10

(6.10)

6.10

(6.10)

106.10

106:10

0 1y 2y 3y


short coupon bond, long 2-year zero, and long 1-year zero—would bring a cash inflowof c¼ 88.8320 today. If the first one—long 3-year zero—costs less than c¼ 88.8320 to enterinto, then the arbitrageur would have a positive cash flow today. Her future obligationswould be matched at every point in time and can be depicted as:

In the future, she would simply collect the receipts from her 1-, 2-, and 3-year zeroinvestments and use them to satisfy her coupon obligations to the lender of the 3-yearcoupon bond. Again, she would not be the only one pursuing this arbitrage strategy. Allinvestors would immediately do the same, driving the price of the new 3-year zero upand its rate down to the 6.10% level. They might also force the yields on the otherbonds to adjust to a level at which arbitrage would not be possible.

2.4 INTEREST RATE RISK

A vast majority of coupon bonds are issued at par or at a price close to par. Practically,what that implies is that at the time of issue the coupon rate is set close to the prevailingmarket yield. Zero-coupon bonds are sold at a discount from par, and the discountedprice reflects the market yield. After the issue, all bond prices fluctuate with marketyields. The coupon rate set at the time of issue does not change and the cash flows itdefines do not change; neither do the dates of the cash flows. But as yields required byinvestors change, so do the discounted values of those cash flows. This phenomenon isnormally portrayed as a downward-sloping concave relationship between the price of abond and the YTM on the bond.Consider a 12-year, 10%, semi-annual coupon bond yielding 10% semi-annually and

let us assume a face value of $100. The bond pays $5 every 6 months for the next 12years and returns the principal of $100 12 years from today. The price of the bondtoday, equal to the discounted value of its cash flows, is equal to:

P ¼ 10=2

ð1þ 0:10=2Þ þ10=2

ð1þ 0:10=2Þ2 þ � � � þ 10=2

ð1þ 0:10=2Þ24 þ100

ð1þ 0:10=2Þ24

¼ 5

ð1þ 0:05Þ þ5

ð1þ 0:05Þ2 þ � � � þ 5

ð1þ 0:05Þ24 þ100

ð1þ 0:05Þ24and, miraculously, to 100! If we set the denominator yield used to discount cash flowsequal to the coupon rate, then the price will always come out equal to par. This can alsobe seen through a recursive argument. Six months prior to maturity (i.e., 111

2years from

Long 3-year zero

Short 3-year coupon

Long 2-year zero

Long 1-year zero

Net

(6.10)

6:10

0

(6.10)

6.10

0

106.10

(106.10)

0

0 1y 2y 3y


now) the bond price is equal to the discounted value of the principal and last couponreceived at maturity, or:

P11:5 ¼ 105

1þ 0:05¼ 100

Twelve months prior to maturity (i.e., 11 years from now) the bond price is equal to thediscounted value of the next coupon and what the bond can be sold for the next period;that is:

P11 ¼ 5þ P11:5

1þ 0:05¼ 105

1þ 0:05¼ 100

and so on.Now suppose that the market yield on a 12-year bond changes to 8% semi-annual.

The new price of the 10% coupon bond is:

P ¼ 5

ð1þ 0:04Þ þ5

ð1þ 0:04Þ2 þ � � � þ 5

ð1þ 0:04Þ24 þ100

ð1þ 0:04Þ24 ¼ 115:25

That is, in order to receive an above-market coupon of 10%, an investor would have topay an above-par price. Conversely, if the market yield on a 12-year bond changes to12% semi-annual, the price will drop to:

P ¼ 5

ð1þ 0:06Þ þ5

ð1þ 0:06Þ2 þ � � � þ 5

ð1þ 0:06Þ24 þ100

ð1þ 0:06Þ24 ¼ 87:45

Plotting the relationship between the price of the bond and its yield, we would getFigure 2.3:

Figure 2.3 Price–yield graph for 12-year, 10%, semi-annual coupon bond.

Note that there is a maximum price for the bond. If the yield drops to 0, then the valueof the bond will be equal to the sum of its cash flows, or:

P ¼ 5

ð1þ 0Þ þ5

ð1þ 0Þ2 þ � � � þ 5

ð1þ 0Þ24 þ100

ð1þ 0Þ24 ¼ 220

The price for the bond approaches 0 asymptotically as the interest rate goes to infinity.Most of the time, we are somewhere in the middle. The graph is a convex (bowed tobelow) curve due to the nature of compound interest. This can also be seen from the


fact that the change in price is not symmetric. When the yield went down by 2%, theprice changed by 15.25; when the yield went up by 2%, the price changed by �12:55.

Observations An increase in market yield lowers the price of the bond; a decrease inmarket yield raises the price of the bond; the price–yield relationship is convex; interestrate fluctuations are a source of price risk for bond holders.

What does the magnitude of that risk depend on? Three factors come to mind:

. Time to maturity—other things being equal, the longer the maturity of the bond, thelarger the price swings are for the same change in interest rates. This can be seen byrepeating our exercise for a 16-year, 10%, semi-annual coupon bond. At a yield of10% semi-annual, the bond prices to par, at 8% to 117.87 and at 12% to 85.92. Achange in the rate used for discounting the cash flows has a greater impact for alonger bond due to the compound interest effect on later cash flows.

. Coupon rate—other things being equal, the lower the coupon rate of the bond, thelarger the price swings are for the same change in interest rates. The intuition issimilar to the time to maturity argument. The principal repayment’s weight in theoverall present value is greater for a low coupon bond than for a high coupon bond.Therefore, a change in the discount rate affects a low coupon bond dispropor-tionately more.

. Coupon frequency—other things being equal, the less frequently the coupons on thebond are paid, the larger the price swings are for the same change in interest rates.Monthly bonds bring the coupons closer to today than annual coupon bonds and,hence, are less sensitive to the discount rate change.

From the above, we can deduce that zero-coupon bonds have the greatest interest raterisk of all bonds with the same maturity, as there is only one cash flow at maturity.Floating-rate bonds have virtually no interest rate sensitivity as their prices always

return to par after coupon payments. Their price may deviate from par on a non-coupondate only to the extent that the first coupon rate that has been set is different from thediscount rate for that first coupon. For example, today’s 9-month (discount) rate may bedifferent from a 12-month rate set 3 months ago for the current 1-year coupon period.All future coupons will be set at ‘‘fair’’ rates and have a present value of 100.To compare the riskiness of bonds with different characteristics (maturities, coupon

rates, and coupon frequencies), we require a metric of interest rate sensitivity. The mostcommonly used interest rate risk metrics are duration and convexity. Both of thesemeasures are local in nature. They are summary statistics that describe the sensitivity ofthe bond’s price to small changes in yields away from the current level. They are notappropriate for large market moves and they are not global in nature: the duration ofthe bond when yields are at 5% can be vastly different from the duration of the samebond when yields are at 8%. Fixed income portfolio managers dealing with hundreds orthousands of bonds with different maturities, coupons, or issuers routinely use durationand convexity measures to select bonds for their portfolios.

Duration

The universe of bonds, even for the same issuer, can be enormous. Imagine the bondsour XYZ GmbH might have outstanding on any given day: a 5-year, semi-annual, 4%


coupon bond; a 714-year, annual, 3% bond with next coupon due in 3 months; a 6-year

zero-coupon; or a 4-year bond whose coupon starts at 3.5% but increases by 0.75% eachyear. How do we choose which bond to invest in or which one to sell out of a portfolio?

Investors compare bonds that have different characteristics by using the notion ofduration. In its original form, as described by Macaulay,5 duration D is defined as thepresent value-weighted average time to the bond’s cash flows. The largest cash flow fora bond is the principal repayment at maturity. But the bond may also provide sub-stantial coupons prior to maturity. The duration measure takes into account the finalrepayment and any interim flows of all cash flows in computing an average time of thecash flows, which is often much shorter than the maturity of the bond. In the averaging,the cash flows are weighted by their contribution to the total present value of the bond.Duration is defined by Macaulay in terms of time or years. We can say that a bond has,for example, a duration of 3.45 years or that another bond has a duration of 7.29 years,and we can compare bonds based on that number. As we will show, this is a veryintuitive measure.

Duration has also another, more interesting interpretation: modified duration is therelative sensitivity of the bond price to a unit yield change. That unit can be 1%, but apreferred unit of yield change is 1 basis point, or 0.01%. For example, if a bond has amodified duration of 6.94 and the yield to maturity increases by 1 basis point, then theprice of the bond will decrease approximately by 6.94 basis points. Modified duration isthus defined as the percentage change in the price divided by the change in yield, or:

ModD ¼ �DP=PDy

As it turns out, duration and modified duration are closely related through the follow-ing formula:

ModD ¼ D

1þ y

n

where n is the number of compounding periods per year for the yield y. Note that thedenominator is close to one, and duration and modified duration are roughly the same.Often traders talk only of one duration, quoting exclusively the modified durationnumbers as these reflect the local riskiness of bonds. But it is important to understandthe intuitive Macaulay meaning of duration.

Let us consider an example of a 6-year, 7%, semi-annual coupon bond yielding 8%.That is, a $100 face value bond pays $3.50 at the end of each 6-month period andreturns the principal of $100 in 6 years. Table 2.5 presents the logic of the Macaulayduration calculation. Columns 1 and 2 contain times and cash flows. Column 3 containsdiscount factors for each cash flow based on the yield of 8% semi-annual. Column 4computes the present value of each cash flow with the sum of all present values at thebottom. The essence of the duration computation is columns 5 and 6. Column 5presents the percentage that the present value of each cash flow represents in thetotal value of the bond. Each percentage can be construed as the weight of that cashflow in the total in today’s dollars. Lastly, column 6 multiplies those weights by thetimes to the cash flows to arrive at a weighted-average time to cash flows.


5 Frederick Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields, and Stock Pricesin the U.S. since 1856, 1938, National Bureau of Economic Research, New York.

Table 2.5 Macaulay duration calculation logic

Maturity in years: 6Coupon: 7.00Yield: 8.00

Time CF DF PV %PV t�%PV

0.5 3.5 0.961 538 3.365 385 3.53 0.017 71 3.5 0.924 556 3.235 947 3.40 0.034 01.5 3.5 0.888 996 3.111 487 3.26 0.049 02 3.5 0.854 804 2.991 815 3.14 0.062 82.5 3.5 0.821 927 2.876 745 3.02 0.075 53 3.5 0.790 315 2.766 101 2.90 0.087 13.5 3.5 0.759 918 2.659 712 2.79 0.097 74 3.5 0.730 690 2.557 416 2.68 0.107 34.5 3.5 0.702 587 2.459 054 2.58 0.116 15 3.5 0.675 564 2.364 475 2.48 0.124 05.5 3.5 0.649 581 2.273 533 2.39 0.131 26 103.5 0.624 597 64.645 79 67.83 4.069 7

Total 95.307 46 100.00 4.972 0

Figure 2.4 is yet another way to present the concept graphically. Each block representsthe percentage of present value recovered through each cash flow. The height of theblock is taken from the appropriate row of column 5.

Figure 2.4 PV weights of bond’s cash flows.

The weighted-average time to the cash flows, using the block heights as the weights, isequal to 4.968. That number is also the approximate percentage change of the pricecorresponding to a 1% change in yield.The Macaulay concept is extremely intuitive. With a little experience, we can guess

bond durations fairly accurately. Here are some heuristics:

. All other things being equal, the longer the maturity, the longer the duration (i.e., theblocks in our graph extend further out and so the weighted time to the repayment islonger).


. The larger the coupon, the shorter the duration (i.e., the higher the coupon blocks,the less weight is assigned to the principal repayment and the smaller the weightedaverage).

. The greater the frequency of the coupons, the shorter the duration (i.e., as moreblocks are closer to today).

All three of these correspond closely to the heuristics behind the interest rate risk ofbonds. Here are two more observations:

. The duration of a zero-coupon bond is equal to its maturity.

. Floating rate bonds have very short durations equal to the next coupon date.

Let us now examine the more practical meaning of duration, that of the interest ratesensitivity applied to our example bond. The 6-year, 7%, semi-annual coupon bondyielding 8% is valued at 95.3075. We computed the Macauley duration to be 4.9720.The modified duration is then:

ModD ¼ 4:9720

1þ 0:08

2

¼ 4:7807

If the yield on the bond were to increase from 8.00% to 8.15% (i.e., by 0.15%, or 15basis points), then the price of the bond should decrease by 4.7807� 0.15, or 0.7171%.Based on the starting value of 95.3075, this translates into a change to 94.6240. Mathe-matically, this can be expressed as:

Pnew ¼ P½1þ ð�ModDÞDy�¼ 95:3075½1þ ð�4:7807Þð0:15Þ� ¼ 94:6240

An exact calculation of the value of the bond assuming an 8.15% yield produces thediscounted value of the bond’s cash flows equal to 94.6270. In the case of a smallchange in the yield, duration was a very good approximation to the change in thevalue of the bond. This would not be so if the yield change considered were large. Inthe extreme case of the yield going down to 0 (i.e., a yield change of 8%), multiplyingthat change by 4.7807 gives the predicted change in the bond price of 38.2456% of thecurrent value, or $36.4509. Duration would predict that the bond price would increaseto $131.7584. Yet we know that at a zero discount rate the value of the bond is equal toa simple sum of its cash flows or $142.

What went wrong? Duration is a local measure. It is the first derivative of the pricewith respect to the yield. Thus, it is a linear approximation based on a line that istangent to a polynomial curve, the true price–yield relationship. This is represented inFigure 2.5.


Figure 2.5 Price–yield relationship.

Later we show how the duration-based linear approximation can be improved with theuse of convexity.Most computer applications do not compute duration the way we presented it in

Table 2.5. Rather, they revalue the bond using the yield a small number of basis pointsabove and a small number of basis points below the current YTM, and then they dividethe change in the bond’s value by the combined size of the yield change ‘‘blip’’. That is,they compute the sensitivity of price to yield directly. Let us denote the value of thebond with a yield blipped up by dy basis points as Pþdy and the value of the bond with ayield blipped down by dy basis points as P�dy. Then duration can be computednumerically by dividing the percentage price change by the total change in yield, or as:

ModD ¼ �ðPþdy � P�dyÞ=P2dy

In our case the two values can be easily arrived at using a financial calculator or aspreadsheet. For example, using the yield of 8.02 we get Pþ2 ¼ 95:216 39, and using theyield of 7.98 we get P�2 ¼ 95:39864. The duration is thus computed as:

ModD ¼ �ð95:216 39� 95:398 64Þ=95:307 462 � 0:0002 ¼ 4:7806

Of course, we could use a smaller yield change or adjust the centering in the numerator.Note that the estimate is off by 0.0001 due to the fact that we used a large blip of 2 basispoints. The smaller the blip used, the smaller the error. We encourage the reader torepeat the calculation by using a 1 basis point or a 0.5 basis point yield change.The numerical procedure of computing duration is very general and works also for

the most complex bonds with embedded options, for which expected cash flows maychange as we vary the yield.


Portfolio duration

Duration is very popular with managers of large bond portfolios. This is due to its onevery attractive property: the duration of a portfolio is equal to the weighted average ofthe durations of individual bonds. The weight for each bond is simply the proportion ofthe portfolio invested in that bond. This property is a direct result of the fact thatdurations are first derivatives of the bond values with respect to yields and that firstderivatives are additive. Let us look at an example.

Consider the following portfolio of bonds:

Investment Investment Coupon Maturity Duration($ million) (%)

400 20 6.50 12 9.54900 45 5.75 10 7.23700 35 5.25 6 4.85

2,000 100 6.86

Duration ¼ 0:20 � 9:54þ 0:45 � 7:23þ 0:35 � 4:85 ¼ 6:859:

The interpretation of the portfolio duration is the same as that for individual bondswith the qualification that the duration of the portfolio is the sensitivity of the value ofthe portfolio with respect to a parallel shift in yield to maturities. In the case of theportfolio represented in the above table, this could translate into the following state-ment: if the YTM on each bond in the portfolio decreased by 7 basis points, then thevalue of the portfolio would increase by 6.859� 7, or 48.013 basis points. In dollars,that is equal to an increase of 0.004 801 3� $2 billion, or $9,602,600. By knowing onestatistic about the portfolio—its duration—the manager can predict the value changefor the entire portfolio very accurately for small changes in yields!

Often, bond managers engage in what is called duration matching, or portfolioimmunization. These terms refer to a conscious selection of bonds to be added to theportfolio in order to reduce the duration of the portfolio to 0 (i.e., to eliminate allinterest rate risk). This is done by selecting the right amount of bonds to be shorted orby buying bonds with negative duration. Sometimes, managers do not attempt toeliminate risk completely, but rather to ‘‘shorten’’ (decrease) or ‘‘lengthen’’ (increase)the duration of a portfolio by reshuffling the allocations to various bonds. Manymanagers of corporate bond portfolios short government bonds with the same durationto eliminate exposure to interest rates, leaving themselves with pure credit spreadexposure. Commercial banks engage in a form of portfolio immunization, by tryingto decrease the duration of their assets (auto and home loan portfolios) and increase theduration of their liabilities (move depositors to long-term certificates of deposit, orCDs).

Note, in the above example, that if the YTMs do not change in parallel (e.g., somechange by 8 basis points and others by 6 basis points), then the estimate based onportfolio duration will be somewhat inaccurate. However, an estimate obtained bysumming the products of the changes in yields for all bond times will have individualdurations that are still very accurate. This is still much easier than revaluing all bonds.


Convexity

Convexity is often used to improve the accuracy of the duration approximation to thechange in value of the bond. It is important to include it in the approximation for:

. Large changes in YTM.

. Bonds whose price–yield relationship is highly non-linear (e.g., bonds with em-bedded options, some mortgage-backed securities).

Convexity is equal to half the second derivative of bond price with respect to the yield,and as such it measures the average rate of change in the slope of the tangent durationline. Numerically, it can be computed as the following difference formula:

C ¼ 1

2� ðPþdy þ P�dy � 2PÞ=P

ðdyÞ2We already have all the ingredients to compute convexity for our example bond. Let usplug the numbers into the formula to get:

C ¼ 1

2� ð95:216 39þ 95:398 64� 2 � 95:307 46Þ=95:307 46

ð0:0002Þ2 ¼ 14:4270

The convexity number measures the average change in the duration per dy basispoints.6 It tells us to what extent the true price–yield curve deviates from the linearapproximation. What we are mostly interested in is in improving that approximation.In order to do that, we need to multiply the convexity by the relevant yield change Dy toobtain the change in the duration over that entire yield change. This may explain thelogic behind the following duration-cum-convexity approximation formula for thebond price change:

Pnew ¼ P½1þ ðModDþ C � DyÞDy�The percentage price change in the bond value per unit of yield comes from twosources: the duration, which for most bonds will underestimate the magnitude of thechange following a straight line, and the convexity that will correct for that under-estimation by reducing the absolute value of the duration. Using the numbers for ourexample bond, we get:

Pnew ¼ 95:307 46½1þ ð�4:7806þ 14:4270 � 0:0015Þ � 0:0015� ¼ 95:6271

We have improved our estimate considerably and are almost spot on! Recall the truevalue of the bond at a yield of 8.15% was 95.6270.Convexity is widely used as a summary statistic to describe large bond portfolios.

Typically, durations and convexities are computed for several possible yield incrementsrelative to today’s level (e.g., �50, �25, 0, þ25, þ50 basis points). It is important toremember, however, that convexities, unlike durations, are not additive and are com-puted by blipping entire portfolios and revaluing all the bonds in them. Just like withdurations, managers engage in immunization strategies with respect to convexities byadding negatively convex bonds or reshuffling portfolio allocations to reduce orincrease the convexity of the overall portfolio.


6 Duration changes continuously between the original value P and the estimated value Pnew corresponding to the yield changeof Dy. The multiplication of the second derivative by 1

2averages the point estimates of convexity per unit of yield over the

entire range of Dy.

Other risk measures

Duration and convexity calculations as described so far assume that the underlyingcash flows of the bond do not change; only the YTM does. Yet the cash flows of bondswith embedded options often change as the yields change (e.g., a ‘‘blip’’ in the yield on acallable bond may trigger a call provision). In those cases, we can compute alternativemeasures of effective duration and convexity where the changed cash flows are explicitlytaken into account when computing the blipped values Pþdy and P�dy. We should,however, bear in mind that those values are not computed through simple discounting,but, rather, with the use of an option-pricing model. As such they take into accountother inputs, the most important of which is the volatility of the yield.

The volatility of the yield enters into the analysis in a different way too. Imagine aportfolio of two corporate bonds both with the same maturity and both trading at par.But one of the bonds has a much higher coupon reflecting the lower credit quality of itsissuer. Is it realistic to assume that the yields on the bonds will move in parallel or is itmore realistic to assume that the riskier bond’s yield will fluctuate proportionatelymore? The volatility of the yield refers precisely to that concept. Computing portfoliodurations may be of little help in this case. Rather, we may prefer to compute individualdurations and scale the assumed yield movements by the respective yield volatilities toarrive at portfolio value change approximations for more realistic yield movements.

Lastly, let us define the concept of the price value of a basis point, which is closelyrelated to duration. Unlike duration, which is expressed in relative terms, the pricevalue of a basis point (PVBP) measures the absolute value of the change in price ofa bond per unit of yield change; that is, it is defined as:

PVBP ¼ �DPDy

and can be approximated by:

PVBP ¼ �ðPþdy � P�dyÞ=10,0002dy

In our bond example, it could be computed as:

PVBP ¼ �ð95:216 39� 95:398 64Þ=10,0002 � 0:0002 ¼ 0:0456

and is defined in dollars. The interpretation of PVBP is that 1 basis point change in theyield causes $0.0456 change in the value of the bond with a face value of $100. For parbonds, modified durations and PVBPs scaled by 1/100 are identical since percentagechanges and absolute value changes are the same if P ¼ 100.

2.5 EQUITY MARKETS MATH

The main difference between the bond math and the stock math is the degree ofuncertainty about the cash flows. For most bonds, coupons and principal repaymentsare guaranteed by the issuer. Unless the issuer defaults, the cash flows are known inadvance. Because of that, we are able to construct very strict relationships between thedifferent bonds of the same issuer and argue, using the arbitrage principle, that related


yields must be certain numbers. We do not have to invoke any assumptions about theissuer’s business, the company’s management style, or the growth prospects for itsportfolio of investments to compute what price we should pay for a bond or whatyield we are earning given the price.This is not the case with stocks. A company is under no obligation to pay dividends.

If and when it returns cash to its shareholders, whether in the form of dividends orcapital gains, is not known in advance. It is hard to discount cash flows if we do notknow them. Even if we assume that we do, then what rate of discount should we apply?The principle of cash flow discounting requires that the rate used reflects the uncer-tainty of the cash flows. Equity holders get paid only after debt holders do. So thediscount rate must be higher than that applied to bonds. What equity holders get paiddepends on the company’s success. Presumably, the greater the potential payoff in itsinvestments, the greater the risk of these projects. Thus the value of the stock willdepend on the subjective growth estimates for the company’s projects. Given thisuncertainty about both the cash flows and the discount rate, we will be able tocompute the rate of return on the stock, knowing its price or the fair value of thestock assuming the discount rate, but not both. In addition, each relative valuationmethod will depend on the assumption about future profitability of the company’sprojects. Given all these uncertainties, the math applied will be much simpler thanthat for bonds, and, paradoxically, complicating it will not make things more accurate.There is one more point of view to bear in mind. The value of the company’s assets is

equal to the value of its equity plus the value of its debt. So if we know the value of allthe assets, why do we have to make profitability assumptions when valuing stocks andnot when valuing bonds? Aren’t the two supposed to add up to a known value?There are two related ways to answer that question. The first, an option-theoretic

point, is that we never really know the value of the assets until the company getsliquidated or acquired. The accounting book value is of very little help; rather, it ismore realistic to view the value of the debt as known. After all, we can discount thevalue of all future debt obligations. The discount rates can be easily gleaned from bondyields and rates quoted on bank debt. In this way, the equity value can be looked on asequivalent to a call option on the company’s assets after creditors have been paid off.That is, it can be written as a payoff for the call with a strike price equal to the value ofthe debt:

Equity ¼ Max½Assets�Debt; 0�

The related second point is that perhaps the debt value is not so well known. After all,when the credit quality of the company declines and bonds get downgraded, the yieldspreads over the risk-free rates can fluctuate and with them so can the value of the debt.Thus, although for risk-free government debt all the discounting machinery is wellestablished, for risky corporate debt it is not so simple. Even if the value of assets isknown, both equity and debt values can fluctuate with the prospects of the company.This point becomes very clear when we consider hybrid securities like convertible debt.It becomes even clearer when we consider companies close to bankruptcy, where thepossibility that debt holders will soon become equity holders is very real. Add to thatthe uncertainty in the value of the assets (how much is the Starbucks brand worth?) andthe point is made.


A dividend discount model

Let us start by assuming that we know the rate that we should apply to discount thecompany’s future cash flows. Perhaps we examined the company’s business plan andcompared it with that of other similar ventures. Perhaps we used the standard capitalasset-pricing model (CAPM)7 which related the discount rate for the stock to the rateon risk-free government obligations and a market risk premium. In any case, we havedetermined the appropriate rate r, which we will assume to remain constant over time.

We want to buy one share of ABC Corp. ABC will pay a dividend of D1 dollars ayear from today and, at that point, we will be able to sell the stock for P1 dollars. Thenthe present value of the cash flows today (i.e., the fair price for the stock, P0) is equal to:

P0 ¼ D1 þ P1

1þ r

D1 could be 0 here. Also note that we may not know the value P1 that we will be able tosell the share for 1 year from today. However, we can argue that tomorrow’s purchaserwill expect to earn a return of r percent on the stock over the following year. In otherwords, he will pay the discounted value of next year’s dividend and sale price; that is:

P1 ¼ D2 þ P2

1þ rToday’s fair value for the share is:

P0 ¼ D1

1þ rþD2 þ P2

ð1þ rÞ2We can extend the argument to a potential purchaser at the end of year 2, and so on, toget a general formula:

P0 ¼ D1

1þ rþ D2

ð1þ rÞ2 þD3

ð1þ rÞ3 þ � � � þDn þ Pn

ð1þ rÞnWe would have gotten the same result if we had assumed that we did not intend to sellthe stock after 1 year, but that we were going to hold it for n years, collect allintervening dividends, if any, and then sell only at the end of year n. That is, theformula does not depend on our, or anyone else’s, holding horizon n. It does notdepend on who will hold the stock over the next n years. It also does not depend onwhether the company pays any dividends at all (i.e., whether any of the Di’s are non-zero) or whether all potential gains come from price appreciation. We can also extendthe argument to infinity to obtain the price as the sum of all future discounted cashflows:

P0 ¼ D1

1þ rþ D2

ð1þ rÞ2 þD3

ð1þ rÞ3 þ � � � þ Dn

ð1þ rÞn þ � � �

Note that if no one believed that the company would ever pay anything, then the pricewould be 0. That is, if everyone agrees that the company will never be profitable, thenno one would invest in it. This is different from believing that the company would neverpay any dividends. After all, value can come from price appreciation only, and we argue


7 All college finance textbooks provide the exposition of the CAPM which relates the required return on the stock to themarket risk premium through the stock’s beta. For example, see Richard A. Brealey, Stewart C. Myers and Alan J. Marcus,Fundamental of Corporate Finance (4th edn), 2004, McGraw-Hill Irwin.

that price appreciation reflects all potential future cash flows from the company. Theowners could always force liquidation after n years to get Pn.ABC’s sister company, ABC-NoGrowth Corp., pays a constant non-zero dividend D

(i.e., Di ¼ D for all i ). The dividend discount equation simplifies to the perpetuityformula:

P0 ¼ D

r

A more attractive sister company, ABC-ConstGrowth Corp., expects its dividends togrow at a constant rate g. The dividend discount equation simplifies to the growingperpetuity formula:

P0 ¼ D1

r� g

Lastly, ABC-NonConstGrowth Corp.’s dividends are uneven for the first n years andgrow at rate g afterwards. We can discount the first n years of dividends individuallyand then apply the constant growth formula, correcting by the discount factor for nyears, by writing:8

P0 ¼ D1

1þ rþ D2

ð1þ rÞ2 þ � � � þ Dn

ð1þ rÞn þ1

ð1þ rÞnDnð1þ gÞ

r� g

Let us look at some numerical examples. We assume that the discount rate investorsapply to the ABC companies is 15%. If ABC-NoGrowth pays a $6 dividend per shareevery year, the fair value for its stock is:

P0 ¼ 6

0:15¼ $40

ABC-ConstGrowth is expected to pay a $6 dividend next year, but its dividends areexpected to increase by 5% every year (i.e., the dividends are expected to be D1 ¼ 6,D2 ¼ 6:30, D3 ¼ 6:615, . . . , D10 ¼ 9:308, . . . , and so on). The fair value for its stockwill be:

P0 ¼ 6

0:15� 0:05¼ $60

For ABC-NonConstGrowth, let us assume that it will pay a constant dividend of $6 forthe first 3 years and then that dividend will grow by 5%. The price of its stock will be:

P0 ¼ 6

1:15þ 6

ð1:15Þ2 þ6

ð1:15Þ3 þ1

ð1:15Þ36 � 1:05

0:15� 0:05¼ $55:12

Implicit in these price calculations is the assumption that each company will be able tomaintain a particular stream of dividends. For example, ABC-NoGrowth will not haveto grow its dividends, while ABC-ConstGrowth will have to be able to maintain the 5%growth in dividends for ever. Does this mean that ABC-ConstGrowth must necessarilybe a ‘‘growth’’ company and ABC-NoGrowth has no growth?So far all we have considered were cash flows distributed to investors in the form of

dividends or capital gains, not company earnings.


8 Analysts often assume a lower discount rate in the perpetuity stage of the model (last term), assuming steady-state growth individends or free cash flows from then on. The rate used is the average for the economy as a whole (i.e., return on the marketportfolio).

Let us think of our ABC companies as manufacturing operations. Imagine bothABC-NoGrowth and ABC-ConstGrowth invest the same amount in plant and equip-ment. It is hard to imagine that ABC-NoGrowth would be able to maintain a steadydividend without replacing and modernizing its equipment. Its earnings before depre-ciation must grow just to keep the dividend constant. Its earnings net of depreciationcan be constant as long as it pays out all of its earnings as dividends. In that case thedividend payout ratio d is 100% and the earnings retention (plowback) ratio is 0%. Thestock price can also be calculated as:

P0 ¼ E1

r

Suppose, instead of paying a dividend next year, ABC-NoGrowth decides that it willreinvest the dividend in existing operations. What must its return on investment (ROI)be just to keep investors equally happy? Supposing the dividend is skipped only once,we can write the discounted value of the company’s cash flows as:

40 ¼ 1

1þ rP1 ¼ 1

1þ r

E1ð1þ ROIÞr

We apply the perpetuity formula 1 year from today and then discount the fair value 1year from today back to today. It is easy to see that the ROI must be equal to thediscount rate investors apply to the company’s cash flows (i.e., ROI ¼ r). It can also beeasily shown that it does not matter for how long the company decides to reinvestearnings, instead of paying them out as dividends. If the company decides to reinvestearnings for 2 years and then pay them out as dividends, then the fair value formulawould simply reflect that:

40 ¼ 1

ð1þ rÞ2 P2 ¼ 1

ð1þ rÞ2E1ð1þ ROIÞ2

r

In particular, if the company decides never to pay any dividends, then it must ensurethat the reinvested earnings always yield a return that is at least equal to the discountrate.

Next we show that the same logic applies to ABC-ConstGrowth, except that it mustensure that its reinvested earnings not only return the discount yield, but also that thereturn on that return grows at rate g! Recall that, even before the company decides toskip dividends, investors expect growth in the distributed cash flows.

Let us consider a scenario where ABC-ConstGrowth decides that it will not pay outits earnings E1 in the form of dividends D1 1 year from now. Instead, it will reinvestthose earnings in projects yielding the return on investment equal to ROI . We wouldlike to use the same valuation trick as before by applying a growing perpetuity formula1 year from today and then discount it by one period back to today. That is, we wouldlike to write:

P0 ¼ 1

1þ rP1 ¼ 1

1þ r

E1ð1þ ROIÞr� g

In order for investors to continue to be willing to pay $60 for the stock, we needROI ¼ r and for the year 2 earnings E2 ¼ E1ð1þ ROIÞ to continue to grow at rate g.This implies that the production base that yielded year 1 earnings E1 must grow at rate rand it must sustain the growth of those earnings E1 at the rate g. But it must also


sustain the growth in the earnings on earnings E1 � ROI at the rate g. The growingperpetuity formula assumes that the entire numerator grows at the rate g for ever.If ABC-ConstGrowth decides to reinvest next year’s earnings of $6, then that re-

investment must be in projects at least as good as the current ones. That is, they mustguarantee not just a return of future value of $6, but also generate future growth in theearnings on those reinvestments at the rate of 5% per year. So, in year 2 the payoutmust be 6ð1þ 0:15Þ, or D2 ¼ $6:90, and then, growing at 5%, it must be D3 ¼ $7:245,D4 ¼ $7:607, and so on.In general, if a company decides not to distribute earnings in the form of dividends,

the projects that it invests these retained earnings in (i.e., new businesses it goes into)must be at least as good as the current base investment (i.e., they must guaranteeadditional perpetual growth at the rate g). If the company always reinvests its earnings,then all future reinvestments must guarantee that growth rate. This argument alsoapplies to partial reinvestments where a fraction equal to the dividend payout ratio dis paid out in the form of dividends and the remainder 1� d is ‘‘plowed back’’.

Observation A growth company is not simply one whose earnings grow, but one forwhich any new investment, whether in the form of retained earnings or additionalcapital, is expected to produce new earnings that will grow at a rate at least equal tothe return on its current earnings.

That is a tall order especially if the current market price already assumes that thegrowth will continue for ever.

Beware of P/E ratios

Price/Earnings (P/E) ratios are a popular metric for valuing stocks. They are intuitive asthey indicate how much per $1 of earnings investors pay in a stock price. Consider twomore ABC sisters. ABC-Value Corp.’s shares trade at $40 and the company is expectedto earn $6 per share. ABC-Growth Corp.’s shares trade at $60 and the company isexpected to earn $6 per share. A P/E enthusiast would compute a P/E ratio for ABC-Value Corp. to be 6.67 (40 divided by 6) and for ABC-Growth Corp. to be 10 (60divided by 6). These could be interpreted to mean that investors pay a lot more perdollar of earnings for the ABC-Growth shares. Should they then sell ABC-Growth andbuy ABC-Value?Simply computing the P/E ratio does not offer any clues as to what is expected to

happen to those earnings. It may be that ABC-Value is simply our ABC-NoGrowthcompany, which replaces its production base only to return a level stream of dividends,while ABC-Growth is our ABC-ConstGrowth company, which, in addition to depre-ciation replacement, offers sustained growth in earnings. It is easy to imagine ABC-Value to be a mature company with stable earnings, but no growth prospects, andABC-Growth as a young entrant into a new, rapidly expanding industry. Why wouldone not invest in the latter? After all, its growth rate may even exceed that imputed in itsshare price, while ABC-Value may fail to deliver the steady earnings it has produced sofar. The situation could easily be reversed: ABC-Growth’s assumed growth may not liveup to its expectations, while ABC-Value may enter growth businesses. What determinesthe value of a stock is not only its current earnings, but also the earnings growth


prospects and the riskiness of the overall business as summarized by its capitalization(discount) rate. None of these factors remain constant and there are many less tangibleones (management, state of industry, economy, etc.).

It is even harder to defend the use of more complicated ratios, like price/earnings/growth (PEG), popularized in the late 1990s during the technology boom. Thesepurported to allow comparisons among companies with tiny current earnings, butenormous ‘‘growth’’ prospects. For these companies, P/E ratios are unappetizinglylarge, but if divided by large growth rates they become supposedly manageable andmeaningful. Let us simply offer an extreme example: How much would you be willing topay for a brilliant idea of a brilliant product in someone’s head?: no earnings, but highpotential growth?

2.6 CURRENCY MARKETS

Currencies are commodities like gold, oil, or wheat. A c¼ 20 note can be, without anyloss of value, replaced by 20 c¼ 1 coins. Normally, the price of commodities is quoted interms of a monetary unit per a commonly used quantity of commodity, like bushel,barrel, or metric ton. We rarely ask the question in reverse: for example, how manybarrels of oil can $1 buy? Foreign exchange (FX) or currency rates are special in thatthey are often quoted both ways: £1 sterling may cost U.S.$1.5 or U.S.$1 may cost£0.6667. Expressing the price in pounds per dollar is equivalent to fixing it to thereciprocal in dollars per pound. The other feature distinguishing currencies fromother commodities is that we often want to know the cross-ratio. We are rarely inter-ested in how many barrels of oil 20 bushels of wheat can buy. However, when returningto the U.K. from a vacation in Mexico, we may want to know how many pounds theleftover 300 pesos will get even though the exchange rate may be quoted in pesos perdollar.

Let us review quote conventions and some potential issues. In what follows, we usethe easily recognizable three-letter currency codes as adopted by the payment clearingsystem SWIFT. Most currencies around the world are quoted with respect to a vehiclecurrency, which is one of the major hard currencies (e.g., USD, GBP, or EUR). Acurrency can be quoted in European terms (i.e., in currency per dollar) or in Americanterms (i.e., dollars per currency). Most former Commonwealth currencies (AUD, NZD,etc.) and the euro follow the latter convention. Most others (e.g., CHF, JPY, HKD)follow the former. Remembering this is only important when observing quotes orpercentage appreciation rates that are not labeled; this text follows the notation inwhich a spot foreign exchange rate X is labeled with the terms in square brackets oras a superscript. The terms describe the ‘‘pricing currency’’ in the numerator and the‘‘priced currency’’ in the denominator. This is analogous to everyday supermarketpricing where £1.50 per 1 loaf of bread can be written as £1.50/loaf. That is, theprice in £/loaf is 1.50. In the same way, the price of 1.5 dollars per pound sterling iswritten as:

XUSD=GBP ¼ X

�USD

GBP

�¼ 1:50

The readers should be aware that confusing notations abound in the market, where acable rate may be written as GBP/USD (i.e., with the base currency coming first)


followed by a number like 1.65 which obviously means 1.65 USD/GBP and not theother way around.For any two currencies, the FX rate in currency 1 per unit of currency 2 uniquely

determines the FX rate in currency 2 per unit of currency 1 (i.e., X ½Curr1=Curr2� ¼1=X½Curr2=Curr1�). In our example:

X

�GBP

USD

�¼ 1

X

�USD

GBP

� ¼ 0:6667

Most FX rates are quoted to four decimal places, except when the whole number islarge, then they are quoted to two decimal places (e.g., a JPY/USD quote may be119.23). Often, the quotation units follow a convention and are dropped. This maylead to misinterpretation of appreciation statistics. Consider the following table whichis similar to many you see in the press:

Currency changes against the USD as of XX/XX/2003

Last Change %Change

AUD 0.5457 �0.0014 �0.3EUR 1.0205 �0.0020 �0.2CHF 1.6844 �0.0030 �0.2

Does this mean that all the listed currencies depreciated against the dollar? On thecontrary, the CHF, quoted as CHF/USD, has actually appreciated as it takes 0.0030fewer Swiss francs to buy $1 than it did yesterday, while the other two currencies, AUDand EUR, may have in fact depreciated.Another confusion comes with the percentage change statistic. If the AUD costs

USD 0.5457 today and it cost 0.5471 yesterday, then we can compute the percentagedepreciation as:

%DXUSD=AUD ¼ �0:0014=0:5471 ¼ �0:002 559 ¼ �0:2559%

as shown in the table. Does this mean that the USD has appreciated by 0.2559%?Absolutely not. Today the USD costs:

XAUD=USD ¼ 1=XAUD=USD ¼ 1=0:5457 ¼ 1:832 509

Yesterday it cost:

XAUD=USD ¼ 1=XAUD=USD ¼ 1=0:5471 ¼ 1:827 819

or AUD 0.004689 more. The percentage change in the value of the USD expressed inAUD is then:

%DXAUD=USD ¼ 0:004 689=1:827 819 ¼ 0:002 566 ¼ 0:2566%

The simple explanation is that the percentage change is not equal to the negative of thepercentage change in the reciprocals. Over longer periods of time the differences can begreater.


Additional complications arise when looking at analysts’ mean forecasts of currencyrates. Suppose that currently:

XUSD=AUD ¼ 0:75, or, equivalently, XAUD=USD ¼ 1 13

Suppose half the analysts polled predict the rate to go to XUSD=AUD ¼ 0:50 (i.e.,XAUD=USD ¼ 2) and the other half to XUSD=AUD ¼ 1:00 (i.e., XAUD=USD ¼ 1). Onaverage, in USD/AUD terms, they predict the rate to be 0.75 (i.e., no appreciation).At the same time, using reciprocals, on average they predict the rate in AUD/USDterms to be 1.5 (i.e., a USD appreciation from 1 1

3). We can come up with examples

where, on average, both currencies may be expected to appreciate at the same time! Themain lesson from these examples is that we always need to be aware of what quotationterms are assumed when interpreting statements about appreciation or depreciation.

The main law governing spot currency trading is the law of one price. In the inter-dealer market, all spot FX rates have to be in line with each other in such a way thatbuying one currency through a vehicle is no cheaper/more expensive than buying itdirectly. The rule, of course, does not apply to retail markets. If we observe in NewYork XJPY=USD ¼ 118:50 and XHKD=USD ¼ 7:80, and at the same moment a dealer inTokyo quotes XJPY=HKD ¼ 15:02, we can profit because in New York 1HKD costs JPY15.192. The law of one price is violated, as the same commodity, the HKD, trades attwo different prices at the same time. We can buy it where it is cheaper and sell it whereit is more expensive.

If we had JPY1,000,000 and used it to purchase HKD 66,577.8961 in Tokyo, then inNew York sold the HKD 66,577.8961 for USD 8,535.6277 and sold the USD8,535.6277 for JPY 1,011,471.88, we would make an instant profit of JPY 11,471.88.This triangular arbitrage would be possible because HKD is cheap in the direct (cross)market in Tokyo relative to the indirect market in New York. Dealers in New Yorkwould sell dollars for yen, convert the yen into HK dollars in Tokyo, and, with HKdollars, buy back US dollars in New York. All dealers would transact in the samedirection. The USD-into-JPY trade would drive up the yen, the JPY-into-HKD tradewould drive up HKD relative to JPY, and the HKD-into-USD trade would drive theUSD up against the HKD. Most likely all three quotes would change until the yen priceof HK dollars is the same in New York and Tokyo. More discussion of triangulararbritrage is provided in Chapter 5.

In all of the above, we ignore the transaction costs in the form of a bid–ask spread. Inreality, we need to modify the computed amounts by considering that we would sell atthe bid and buy at the ask. Instead of simply one FX cross-rate, we need to compute itsbid and ask. The principle remains the same.

The law of one price binds hundreds of possible currency combinations together. Assoon as one FX quote changes in the market, many others follow.


___________________________________________________________________________________________________________________________________________________________________________ 3 __________________________________________________________________________________________________________________________________________________________________________

___________________________________________________________________________________ Fixed Income Securities ___________________________________________________________________________________

To finance their activities, governments, financial, and non-financial corporations raisedebt funds by borrowing from financial institutions, like banks, or by issuing securitiesin the financial markets. Securities are distributed in the primary markets, where theyare sold directly from borrowers to investors, sometimes with the help of an investmentbank. They are traded among investors in the secondary markets. Securities marketscan be, in general, divided into money and capital markets. Money market instrumentsare those whose maturities are less than 1 year. Capital market instruments are thosewhose maturities are more than 1 year; they include preferred and common stockswhose maturities are infinite. This division is largely artificial and due to differentlegal requirements.In this chapter, we review the spot markets for debt securities, also called fixed

income securities. Debt contracts typically have a stated maturity date and pay interestdefined through a coupon rate or a coupon formula. They include a variety of moneymarket instruments and long-maturity securities, like straight bonds, asset-backed debt,and mortgage-backed securities. They also include spot-starting swaps and other non-securities (Chapter 8 covers swaps in detail). These may be private derivative contracts,but because of their fungible nature are frequently created simultaneously with bonds inorder to change the nature of the issuer’s liabilities.The review of each product is brief. We outline the main features of each instrument

and the structure of the market. We provide recent growth statistics with breakdownsby types, currencies, etc., to provide a feel for the richness of the markets. For productdetails, readers are referred to the many voluminous ‘‘handbooks’’ available in book-stores. For market statistics, readers are referred to the publications of the U.S. FederalReserve, Bank for International Settlements, International Swaps Dealers Association,rating agencies, and other industry sources.

3.1 MONEY MARKETS

Money market securities are debt instruments of high credit quality issuers withmaturities up to 1 year. They have short durations, low convexities, and very lowdefault probabilities. Most are issued by governments, and prominent financial andnon-financial corporations. They are very liquid. They trade in large denominations inthe over-the-counter (OTC) markets with many buyers and sellers present at all times.We describe the markets for the following instruments: U.S. Treasury Bills (T-Bills) andU.S. federal agency discount notes, short-term municipal securities, Fed Funds (U.S.),bank overnight refinancing (Europe), repurchase agreements (repos), Eurocurrency

deposits, commercial paper, and other (negotiable certificates of deposit, or CDs,banker’s acceptances).

U.S. Treasury Bills

About one-fifth of all U.S. government marketable debt is in the form of T-Bills, whichare book-entry (no paper security) discount (zero-coupon, issued at a discount frompar) securities issued by the U.S. Treasury and initially distributed by a handful ofprimary dealers who participate in auctions conducted by the New York FederalReserve Bank, similar in format to those for Treasury notes and bonds. They tradein the most active and liquid market (i.e., they can be easily sold and bought prior tomaturity).

In the secondary market, T-Bills are not quoted in terms of prices, nor are theyquoted in terms of meaningful yields. The quoting convention they follow, the Act/360 discount yield basis, derives its logic from percentage rebates of list prices and ismost unappealing. We review this bizarre code.

On April 23, 2003 some T-Bill prices were quoted in the Wall Street Journal asfollows:

The bid/asked discount yields of 1.13/1.12 quoted on the June 19 T-Bill are not realyields, but strange shorthand for prices. Normally, we would want to compute a rawyield earned on a security as ðPend � PbeginÞ=Pbegin, where Pbegin is the purchase price andPend is a redemption price. Unfortunately, the T-Bill convention assumes somethingmore akin to: ðPend � PbeginÞ=Pend . T-Bills are sold at a discount from par, and soPend ¼ 100. The annualized discount yield is defined as:

ydisc ¼ ð100� PÞ100

� 360Act

where Act represents the actual number of days to maturity. So, if we know thematurity of the T-Bill and the quoted discount yield ydisc, we can figure out the priceby solving for:

P ¼ 100� 100 � ydisc � Act360

Using the asked discount yield for the June 19 T-Bill, this translates into:

P ¼ 100� 100 � 0:0112 � 57=360 ¼ 99:8227

The T-Bill is offered for a price of 99.8227% of the face value. Once we have the price,

MATURITY

May 15 03Jun 19 03

BID

1.121.13

ASKED

1.111.12

CHG

. . .�0.01

ASKYLD

1.131.14

DAYS TOMAT

2257


we can compute the real yield (on an Act/365 basis) on a purchase of the T-Bill. Wedivide the numerator by the purchase price Pbegin instead of the sale price Pend :

y ¼ 100� P

P� 365Act

¼ 100� 99:8227

99:8227� 36557

¼ 0:0114 ¼ 1:14%

The result (1.14%) is what is reported in the last column as the ask yield.Many money market instruments, which we review next, follow the same discount

yield-based rather than price-based quotation convention.

Federal agency discount notes

The U.S. government has sponsored several agencies whose job is to pursue publicpolicy goals that include housing, education, or farming support. The agencies areset up as governmental units or public corporations. A few enjoy full, and mostenjoy implicit credit guarantee of the U.S. government.1 The mortgage supportagencies, like Federal Home Loan Banks, the Federal Home Loan MortgageCorporation (Freddie Mac), the Federal National Mortgage Association (FannieMae), or the Government National Mortgage Association (Ginnie Mae), issue theirown short- and long-term debt obligations and buy, from commercial banks, mortgageloans that conform to certain standardized norms. This replenishes the banks’ fundsand allows them to issue new loans. The agencies also sell to investors mortgage-backedbonds in various forms in order to reduce their balance sheets. The Student LoanMarketing Association (Sallie Mae) and the Farm Credit Banks perform similaruseful functions for the student loan and farming markets.The agencies often seek short-term funding by issuing agency discount notes (discos).

These notes resemble T-Bills: they are quoted the same way using discount yields, theyhave similar maturities, and they are sold at a discount from par. Today, there are over$100 billion of these notes sold through a small circle of U.S. broker-dealers. Mostinvestors hold them to maturity ensuring nearly risk-free return. The trading in them isthin with the spreads above T-Bill yields reflecting both the relative lack of liquidity andminimal credit risk. The main buyers are money market mutual funds and commercialbanks.

Short-term munis

State and local governments in the U.S. issue short-term securities of two kinds:interest-bearing and discount notes. The interest-bearing notes typically have variablerates tied to T-Bills. The spreads over T-Bills on both types are negative to reflect theirtax-exempt status. Like their long-term counterparts, both can be unsecured generalobligation securities (GOs are backed by the credit of the municipality) or revenuesecurities (backed by revenue from specific projects). Revenue paper is generallyriskier. Trading is thin (most are held to maturity) and credit varies greatly.

Fixed Income Securities 69

1 Investors perceive the U.S. government unlikely to let such large and important institutions fail. Thus they enjoy a lowercost of funding than comparable private banking conglomerates. Recently (e.g., see the Wall Street Journal of February 25,2004), some members of Congress as well as the Chairman of the U.S. Fed, Alan Greenspan, have recognized that the largestagencies pose serious financial risks to the system and urged curbs on their growth and separation from the credit backing ofthe government.

In Europe, regional authorities, state-run organizations, and sometimes cities issueshort- and long-term bonds. They do rely, to a greater extent than their U.S. counter-parts, on intermediaries and typically negotiate loans directly from banks.

Fed Funds (U.S.) and bank overnight refinancing (Europe)

Despite its name, the Fed Funds (FFs) market is administered solely by private finan-cial institutions and is largely unregulated by the U.S. government. The core of themarket has been the trading of excess reserves by commercial banks that must satisfyreserve requirements set by the central bank against their deposits. Banks with excessreserves lend them as overnight funds to banks short of cash reserves. The marketcomprises any funds in reserve accounts at the Federal Reserve Bank. These are usednot only to satisfy the stated reserve ratio, but also to clear securities and wire transac-tions in the U.S. These include purchases and sales of commercial paper, banker’sacceptances, T-Bills, etc., as well as U.S. Treasury bonds and notes. Settlementoccurs through an immediate transfer of funds that can be used to effect other transac-tions the same day.

Trading in FFs involves overnight borrowing and lending of claims on cash depos-ited at the Federal Reserve. A borrower of cash is called an FF buyer, a lender is an FFseller. It is an interbank market; only commercial banks are permitted to hold balancesat the Fed. Over 10% of the daily volume is attributed to banks acting on behalf of non-banks. Banks act as dealers and trade FFs either directly (� 60%) or through brokers(� 40%), like Prebon, Garvin, or Noonan. FF loans are unsecured; there is no protec-tion against default. About 75% of transactions are overnight, the rest are term FFs ofup to 6 months.

The FF rate is set by the transacting banks. However, it is also the main tool used bythe central bank to control the money supply. Cash owned by banks is subject to amoney multiplier in the process of credit creation. The Federal Reserve, not being aparty to any of the FF transactions, attempts to affect the FF rate through FederalOpen Market Committee (FOMC) announcements, followed by open market opera-tions. In their eight meetings throughout the year, the Fed governors state their pre-ferred level for the FF rate, and on the next day back their words with actions bydirecting the manager of the open market operations desk in New York to buy or sellsecurities for cash, thereby injecting or removing cash from the economy. These secur-ities transactions are done outright (permanent) or as system repos (temporary). Mostof the time, the mere statement from the Fed alters the rate banks charge each other.But it is worthwhile remembering that the real influence is indirect through open marketsecurities transactions. On rare occasions, like in the days after September 11, 2001, inorder to ensure liquidity the Fed flooded the market with cash (in effect causing the FFrate to drop dramatically) by buying securities.

In Europe, the European System of Central Banks (ESCB) conducts refinancingoperations. Banks trade funds to satisfy reserve requirements, charging each other anovernight rate. The volume-weighted average of all overnight unsecured lending trans-actions initiated within the Eurozone by a particular panel of banks (with the highestvolume) is known as Eonia.


Repos (RPs)

Legally, repos, or repurchase agreements (RPs), are simultaneous spot sales of market-able securities and (forward) repurchases of the same securities at a pre-specified (higher)price at a later date. Economically, they are borrowings: a sale results in a cash inflowand the subsequent repurchase in a larger cash outflow. The difference is the interest onthis collateralized loan. The arrangement is typically over-collateralized through anupfront haircut. That is, the amount borrowed is less than the market value of thecollateral. The borrower (i.e., the seller/repurchaser) is said to enter a repo; the partyto the other side of the transaction (i.e., the buyer/forward reseller or the lender) is saidto be enter a reverse repo. The interest rate is quoted as simple interest on an Act/360basis. Since the collateral is composed of marketable securities (Treasuries in the U.S.,government bonds in Europe), the biggest users of the repo market, on both sides, arebroker-dealers (followed by banks) as they have access to the collateral. For broker-dealers, the repo market is the primary source of funds ( just like FFs for banks). Centralbanks use repo markets to conduct monetary policy through open market operation.Municipal governments with seasonal cash flows use repo markets to temporarily investtheir cash balances in credit risk-free assets. Repos can be overnight or term with thelongest reverse maturity dates up to 1 year. Sometimes the repo is open, which means it isautomatically renewed every day until one party decides to close it down.For broker-dealer firms, repos are the primary means of financing the inventory and

covering short positions. But almost every dealer firm also uses the repo market to run amatched book, by taking on repos and reverse repos with the same maturity, andcapturing the spread between the funds lent (reverse repos with non-dealers) andborrowed (repos with dealers and non-dealers). The principle at work here is simplearbitrage of lending at a higher rate than borrowing. Theoretically, the collateral in amatched book should be of the same quality on both sides; practically, part of thespread may come from mismatched collateral. Dealers often use repo brokers to identifyparties willing to enter reverse repos (from the dealer’s perspective).The biggest repo market by far is in the U.S, but the European repo market has

grown considerably. It is close to EUR 3.4 billion, with 88.8% collateralized by govern-ment bonds and over 50% being cross-border transactions, as of the end of 2002,according to the International Securities Market Association (ISMA).

Table 3.1 European RPs

By area By currency By collateral(%) (%) (%)

Domestic 42.9 EUR 77.2 Germany 28.9Eurozone 24.0 GBP 10.0 Italy 18.5Non-Eurozone 26.4 USD 7.7 France 10.7Anonymous ATS 6.7 DDK, SEK 2.0 Spain 6.9

JPY 2.2 Belgium 5.3Other 0.8 Other Eurozone 5.3

U.K. 10.8DKK, SEK 2.3U.S. 2.6Other 7.7Unknown 0.9

Source: ISMA, European Repo Market Survey, December, 2002.


More than 75% of the Euro RPs are denominated in euros, with close to 30% of thecollateral coming from Germany. Close to 40% of repos are arranged directly betweentwo parties and slightly less brokered by repo brokers. Legally, more than 75% of theEuro market are standard repos and close to 90% of them have a fixed rate.


By arrangement By contract type By collateral(%) (%) (%)

Direct bilateral 39.5 Classic repo 79.5 Fixed rate 89.7Voice-brokered 36.5 Documented sell/buyback 10.8 Floating rate 7.0ATS 16.8 Undocumented sell/buyback 9.7 Open 3.3Direct tri-party 7.3


Historically, most repos have been overnight. Recently, term repos with maturitites upto 1 month have gained prominence, accounting for around 50% of the market. Of theoverall lending activities collateralized by financial assets, RPs represent over 85% ofvolume. The market is moderately concentrated with the top 10 dealers accounting forabout 50% of transactions and the top 30 dealers accounting for close to 90% oftransactions.


Lending collateralized by financial assets RP Concetration(%) (%)

Repo 86.4 Top ten dealers 50.9Securities lending 13.6 Top 11–20 22.1

Top 21–30 14.6Rest 12.4


Eurodollars and Eurocurrencies

A Eurocurrency is a currency deposited to earn interest outside its country oforigin. U.S. dollars held in banks outside of the U.S. (not necessarily in Europe) arecalled Eurodollars (EDs). Euroeuros are euro deposits outside the Eurozone (say, inJapan). The main center of activity is London. The size of the market is largelyunknown, but estimated at over $2 trillion outstanding (known to be greater thanany other money market), and dominated by EDs. Based on the volume of futurestrading, the ED represents close to 90% of the daily transaction activity of the entiremoney market.


Figure 3.1 U.S. Treasury and private instruments in the dollar money market, daily averagetransactions (in billions of U.S. dollars and percentages): (a) T-Bill and Eurodollar futurestransactions; (b) Eurodollar turnover as a percentage of money market activity (including cashmarket transactions in T-Bills).Data from Federal Reserve Bank of New York (FRBNY), FOW TRADE, and BIS calculations. Source: R. N. McCauley,

Benchmark tipping in the money and bond markets, BIS Quarterly Review, March, 2001. Reproduced with permission from

Bank for International Settlements.

Since Eurocurrency transactions exist outside the authority of any national govern-ment, they are free from regulation and intervention.2 The origin of the market datesback to the Soviet deposits of U.S. dollars in Paris and London (rather than in the U.S.)in the 1950s, Arab deposits of dollars in Europe after the Suez War of 1956 and therecycling of ‘‘petrodollars’’ in the 1970s. The U.S. and Japan allowed their domesticbanks to participate in the ED and Euroyen markets through International BankingFacilities (IBFs) and Japanese offshore markets, respectively. Today, the Eurocurrencymarket is the foundation of all swap and all Eurobond activity, the scene of the greatestinnovations in finance. Free of regulation, deposits are traded actively around the clockwith ED and other Eurocurrency futures markets alternating between London,Chicago, and Singapore. The spot deposit rate, quoted on an Act/360 basis fordollar deposits, is determined daily by a rate fixing of the most active London banksat 11 a.m. GMT.3 The settlement is same-day through a clearing system called CHIPSwhich competes with the same-day settlement procedure of the Fedwire. Swaps basedon EDs commonly take two London business days to settle (e.g., a 90-day deposit oftoday would start earning interest 2 days from today and end earning interest 92 daysfrom today). For spot deposits (i.e., placed today), the maximum maturity tends to be12 months with the most common one being 3 months, for all currencies. Many broker-dealers, money center banks, and large institutional investors participate in boththe spot and forwards market on both sides (i.e., as lenders and borrowers). In the

(a) (b)


2 Strictly speaking, governments do regulate the banking facilities engaged in Eurocurrencies. However, at a minimum,Eurocurrency deposits are not subject to reserve requirements and deposit insurance fees.3 It is officially calculated by the British Bankers Association (BBA).

institutional market, deposits can be arranged as time (fixed maturity), placements(overnight), or as call money (no maturity, withdrawn at will).

The spread between the 3-month ED deposit rate and the T-Bill rate is known as theTED spread. It reflects the difference in the default probability of the average creditquality of ED market participants and the risk-free rate (the same way the swap spreaddoes). Presumably, it also reflects the supply and demand for unrestricted deposits (i.e.,the relative demand for borrowing vs. lending). The TED spread varies over time andspikes up in times of economic uncertainty.

Figure 3.2 Spread between U.S. Treasury and private yields (in basis points). The TED spread ismeasured as the monthly average of the spread between the 3-month T-Bill and ED rates.Data from Datastream and BIS calculations. Source: R. N. McCauley, Benchmark tipping in the money and bond markets,

BIS Quarterly Review, March, 2001. Reproduced with permission from Bank for International Settlements.

Negotiable CDs

Certificates of deposit (CDs) similar to those offered to retail customers are tradedamong institutional investors in most major currencies. Banks post daily currentrates on an Act/360 basis for denominations larger than the equivalent of $100,000.Institutional investors typically buy them at a discount from par and can trade themprior to maturity, though they rarely do.

Bankers’ acceptances (BAs)

BAs are a popular form of export and import finance. They originate when banksguarantee time drafts open on behalf of importers who promise to pay for goods ontheir receipt. Exporters often sell them at a discount prior to their due date and theybecome de facto obligations of the guaranteeing bank.

Commercial paper (CP)

CP represents unsecured promissory notes issued in large denominations by large, well-known corporations, with maturities ranging from a few days to 9 months. These are


sold directly to investors (direct placement paper) or through dealers (dealer-directedpaper). Issuers must be sufficiently known to the investing public to access the marketdirectly. Direct paper is typically sold by the salesforce of financing arms or financesubsidiaries of large corporations, like GE Capital or GMAC in the U.S. The 9-monthmaturity limit is an artifact of a U.S. registration exemption. CP represents about 25%of all commercial and industrial lending in the developed world. Significantly more thanhalf of the CP is issued in the U.S. where the secondary market is not active. The EuroCP market maturities are shorter (up to 180 days), but more active in secondarytrading. Foreign issuers favor the U.S. domestic market where they constitute about50% over the ED markets where the interest rates are slightly higher. Financial regula-tions in parts of Europe make it less attractive or impossible for foreign borrowers toissue paper. The volume of offerings tends to be cyclical. The largest holders of CP areretail money market mutual funds. CP is quoted on a discount yield basis (360-dayyear) similar to U.S. T-Bills.Until the 1980s, there were only three CP markets: the U.S., Canada, and Australia.

The U.S. CP still accounts for nearly 77% of the global CP outstanding, with 50–75%of new issuance. Net issuance fluctuates dramatically year to year.

Figure 3.3 Domestic and international net issuance of commercial paper (in billions of U.S.dollars). Data on domestic issuance for the first quarter of 2001 are preliminary.Data from Euroclear and national authorities. Source: B. H. Cohen and E. M. Remolona, Overview: Are markets looking

beyond the slowdown? BIS Quarterly Review, June, 2001. Reproduced with permission from Bank for International

Settlements.

According to the U.S. Federal Reserve, as of August, 2000 the global outstandingstotaled $1,538 billion, around 40% of that being asset-backed CP. The dynamicgrowth in the issuance of asset-backed CP (collateralized by revenue from specificassets) is a relatively recent phenomenon which has been brought about by the thirstfor low-cost, short-term capital and the entry into the CP market of lower credit qualityissuers.


Figure 3.4 Exhibit 1—monthly global CP outstandings, January, 1994–August, 2000, outstand-ings double in 5 years amid rising interest rates.Data from Board of Governors of the Federal Reserve System. Source: Global Credit Research, October, 2000 and

Commercial Paper Defaults and Rating Transitions, 1972–2000. Copyright Moody’s Investors Service, Inc. and/or its

affiliates. Reprinted with permission. All rights reserved.

The Euro CP market nearly tripled in the latter part of the 1990s to about $268 billionby 2000. It continues to enjoy extraordinary growth. The third largest, the Japanese CPmarket, stagnated by the end of the 1990s to the equivalent of about $152 billion.Figures 3.5–3.7 show the growth of CP in Europe, non-U.S., and nascent markets.

Figure 3.5 Exhibit 2—Euro CP outstandings, market nearly triples in 5 years.Data from Bank for International Settlements. Source: Global Credit Research, October, 2000 and Commercial Paper

Defaults and Rating Transitions, 1972–2000. Copyright Moody’s Investors Service, Inc. and/or its affiliates. Reprinted

with permission. All rights reserved.


Figure 3.6 Exhibit 3—CP outstandings by country in major non-U.S. markets, worldwide CPmarkets generally enjoy steady growth.Source: Global Credit Research, October, 2000 and Commercial Paper Defaults and Rating Transitions, 1972–2000.

Figure 3.7 Exhibit 4—CP outstandings by country in Nascent markets, European and LatinAmerican entrants spur growth. Outstandings for 2000 are preliminary estimates.Source: Global Credit Research, October, 2000 and Commercial Paper Defaults and Rating Transitions, 1972–2000. Copyright

Moody’s Investors Service, Inc. and/or its affiliates. Reprinted with permission. All rights reserved.

About 35% of the CP is issued by non-bank financial institutions. They are followed byindustrial corporations (17%) and banks (12%)

Figure 3.8 Exhibit 5—Moody’s rated CP by sector (as of August 1, 2000; 2,071 ¼ 100%), bankand non-bank financial institutions dominate market. ‘‘Other’’ segment includes retail, media,sovereign, transportation and hotel, gaming and leisure.Source: Global Credit Research, October, 2000 and Commercial Paper Defaults and Rating Transitions, 1972–2000. Copyright

Moody’s Investors Service, Inc. and/or its affiliates. Reprinted with permission. All rights reserved.

Close to half of Euro CP is unrated, whereas most issues in the U.S. are rated either byS&P or Moody’s. In Europe there are fewer rating categories, ranging from prime 1 toprime 3 for investment grade issues and non-prime for non-investment grade. Thecorrespondence to Moody’s bond ratings is shown in Figure 3.9.


Figure 3.9 Exhibit 6—Moody’s short-term vs. long-term ratings.Source: Global Credit Research, October, 2000 and Commercial Paper Defaults and Rating Transitions, 1972–2000.

Reproduced with permission from Bank for International Settlements.

At the time of issuance, 80% of CP is rated prime 1 and 17% is rated prime 2. As issuesage (i.e., approach maturity) the probability of default decreases, resulting in the factthat the vast majority (88.9%) of outstanding CP is rated prime 1. No more than 11issuers per year have ever defaulted prior to the year 2000. The frequency of defaults bycountry and industry reflects the composition of the CP market. The U.S., Sweden, andMexico-domiciled issuers lead the country default table, while financials, industry, andutility issuers lead the sector default category.

The spread between prime 1- and prime 2-rated issues has remained relatively stableover time. In general, it has followed the economic business cycle, spiking up dramatic-ally during economic downturns. The frequency of downgrades has been an excellentleading indicator of increases in CP defaults.

Figure 3.10 U.S. CP spreads (30-day yields, in basis points, at month-end): (a) spreads overLIBOR; (b) relative spread.Data from Bloomberg, Datastream, and BIS calculations. Source: B. H. Cohen and E. M. Remolona, Overview: Are markets

looking beyond the slowdown? BIS Quarterly Review, June, 2001. Reproduced with permission from Bank for International

Settlements.


The credit quality of CP relative to EDs depends on the ratings of the CP issues orissuers relative to the average credit quality of the interbank participants of the EDmarket, believed to be approximately AA. The spread over LIBOR (London interbankoffered rate) of prime 1 issuers is normally negative, reflecting their high quality. Thespread of prime 2 issuers is positive. Both spreads are somewhat volatile over time,reflecting changes in the overall economic conditions relative to those of money centerbanks. In the early 2000s, spreads on lower quality CP shot up as many issuers faced adifficult short-term financing environment that forced them to resort to bank financing.This caused the divergence of the perceived default probabilities between the A1 and A2categories.

3.2 CAPITAL MARKETS: BONDS

The global bond market is enormous. The total amount outstanding is close to $40trillion with three-quarters of that amount in domestic and one-quarter in internationalbond markets. U.S. and European financial institutions are the biggest issuers in bothsegments. In recent years, the net issuance (i.e., the excess of the amount issued over theamount retired) has reached close to $3 trillion. Two-thirds of that amount is issued indomestic bond markets and one-third in international bond markets. Governments arethe biggest issuers in both domestic and international markets, representing over one-half of all the new issues.The U.S. and Japanese governments issue almost exclusively in domestic markets,

while emerging markets governments and international organizations issue pre-dominantly in international markets. European governments issue in both. The totalindebtedness (public and private) of U.S. and European residents continues to grow.The U.S. experienced a brief reduction in debt in the late 1990s, but with new govern-ment deficits in the early 2000s debt issuance returned to its previous levels. Domesticprivate debt issuance exploded in the U.S. in the late 1990s, with non-bank financialinstitutions and corporations leading the charge and more than twice offsetting anyreductions in government debt. In Europe, banks and telecom corporations did most ofthe borrowing. In the 1990s, Europeans became the keenest to issue internationalprivate debt, overtaking the U.S. in the market for international bonds. This was theresult of further integration of European capital markets and an explosion of cross-border issuance. The issuance of international bonds by U.S. residents also increaseddramatically.Japan’s net issuance has been almost flat in recent years. Banks retired debts and

corporations added little domestic bond financing to their balance sheets. The pace ofissuance by the government of Japan has, however, accelerated, overtaking the U.S.and Europe to become the largest government issuer in the world.During the same time, emerging economies experienced continued development of

their securities markets. East Asian countries worked through their debt burdens. Bankfinancing in Asia declined, while debt securities issuance remained flat. Latin Americaincreased its indebtedness to the rest of the world, mostly through restructuring of oldand the issuance of new debt securities. The emerging economies of Europe continuedto experience capital inflows. Government and private debt markets continued toabsorb a portion of that inflow.Figures 3.11–3.14 contain statistics for domestic and international bond markets.



(a)

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Figure 3.11 Domestic debt securities—amounts outstanding: (a) all issuers; (b) financial institu-tions; (c) corporate issuers; (d) governments.Source: BIS securities statistics (see http://www.bis.org/publ/qcsv0306/anx....csv with file numbers 12 through 16).

Reproduced with permission from Bank for International Settlements.

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Figure 3.12 International debt securities by nationality of issuer—amounts outstanding; (a) allissuers; (b) financial institutions; (c) corporate issuers; (d) governments.Source: BIS securities statistics (see http://www.bis.org/publ/qcsv0306/anx....csv with file numbers 12 through 16. Reproduced

with permission from Bank for International Settlements).

(c)

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Figure 3.13 Net issuance of domestic and international bonds and notes (in billions of U.S.dollars), by country of residence: (a) U.S.; (b) Europe (EU 15 countries, Norway, Switzerland,and Turkey); (c) Japan.Source: R. McCauley and E. Remolona, Special feature: Size and liquidity of government bond markets, BIS Quarterly

Review, November, 2000. Reproduced with permission from Bank for International Settlements.


Figure 3.14 International bank and securities financing in emerging economies (in billions ofU.S. dollars): (a) Asia and Pacific; (b) Latin America and Caribbean; (c) Europe. Data from Bankof England, Capital DATA, Euroclear, ISMA, Thomson financial securities data, national data,BIS locational banking statistics.Source: B. H. Cohen and E. M. Remolona, Overview: Are markets looking beyond the slowdown? BIS Quarterly Review,

June, 2001. Reproduced with permission from Bank for International Settlements.

U.S. government and agency bonds

The U.S. Treasury has over $4 trillion in total debt outstanding, of which $2.3 ismarketable and actively traded in an OTC market. The securities are considereddefault-free (the government can print money to pay off debt). The only entity thatcan jeopardize that status is the U.S. Congress itself, which it did in 1995 when itrefused to issue new debt to retire the old. More than two-thirds of the newly issuedTreasuries are bought by non-U.S. residents. The Treasury market serves as therisk-free benchmark to other markets (corporate bonds, swaps, etc.). Unlike T-Bills,Treasury notes, with maturities of 2–7 years, and bonds, with maturities of 10–30 years,are quoted in terms of prices with 1/32 as the tick (supplemented with a ‘‘þ’’ as the half-tick 1/64). The whole number is separated from the /32 fraction by a colon or a dash(e.g., 100:27, or 100-27), but sometimes misleadingly by a decimal (100.27). Prices arecommonly converted to semi-annual bond equivalent yields on an Act/356 basis.Current benchmark issues, closest to the 2-, 5-, 10- and 30-year maturities that aremost actively traded, are called on-the-run Treasuries and are normally boldfaced.Treasury notes and bonds pay semi-annual coupon interest. Treasury STRIPs(separated coupons or principals off original Treasuries sold as individual securities,or separate trading of registered interest and principal of securities) trade as zero-coupon instruments sold at a discount from par. On April 23, 2003 some U.S.Treasuries and STRIPs were quoted in the Wall Street Journal as follows:

Government Bonds & Notes

RATE4.2505.375

U.S. Treasury STRIPSMATURITYFeb 09

MATURITYMO/YRNov 03nFeb 31

TYPEci

BID101:21107:04

BID82:11

ASKED101:22107:05

ASKED82:14

CHG�1�3

CHG�1

ASKYLD1.194.90

ASKYLD3.35


The T-Bond paying 5.375% coupon semi-annually (i.e., 2.6875% every 6 months) andmaturing in February of 2031 is offered at a price equal to 107 5

32 percent of par to yield4.90% semi-annual Act/365 equivalent.

At any given time there may be well over a hundred different old and new issues beingquoted and traded. The on-the-run issues comprise the bulk of the trading and arereferred to simply by maturity (e.g., the 2-year or the 10-year). Off-the-runs are referredto by stating the month and year of maturity and the coupon rate. The issues con-sidered current (on-the-run) change as new notes and bonds are sold by the U.S.Treasury following an auction calendar published well in advance. In fact, new on-the-runs start trading even a few days before they are auctioned for the first time (i.e.,even before the coupon rate is known). They trade on a yield basis and appear on dealerscreens as when issued (WI) securities. As soon as they are auctioned (and the couponrate is known) and dealers have them in their inventory, the yield is converted to price.This process allows a smooth repositioning by hedgers (e.g., swap traders) from oldissues into new ones. Hedgers are often balanced. Some buy new issues; some shortthem. Long-term investors (e.g., insurance companies or bond funds) typically buy upold issues to hold to maturity. The comparison of the yields on the WIs and old on-the-runs is the first indication of whether the auction may be over- or undersubscribed. AWI is said to trade rich if the yield on it is lower (future price higher) than that on therelevant current on-the-run.

The U.S. Treasury publishes a schedule (calendar and amounts) of auctions for allnew notes and bonds. Two types of bids are accepted: competitive (large, based onprice/yield) and non-competitive (based on quantity limited to $1 million). At eachauction, winning competitive bidders do not pay the price they bid, but the cutoffstop price arrived at by subtracting from the total the amount of non-competitivebids and competitive bids arranged from the lowest to the highest yield until allbonds are sold. This is called a single-price Dutch auction.

As described in Chapter 2, coupon securities are packages of zero-coupon securities.U.S. Treasuries are packages of STRIPs. Mechanically, STRIPs are created by separat-ing all coupon payments from notes and bonds. For example, 21 STRIPs can be createdfrom a 10-year semi-annual bond: 20 coupon payments and principal. They are thentraded as separate securities with their own CUSIP (identification) numbers. Yields oncoupon bonds and STRIPs are bound by a strict arbitrage relationship.

Let us review these on a simplified example with perfectly even semi-annual periods.Suppose you observe that the 3-year 5% note has a yield to maturity (YTM) of5.3653% (i.e., its price is 99.00). Suppose also that the 6-, 12-, and 18-month STRIPsyield 5.25%, and the 30- and 36-month STRIPs yield 5.37%. Can the 24-monthSTRIPs yield 5.5%? No, its yield will be given by the equation for the price of thecoupon bond with coupons discounted at the zero rates; that is:

99 ¼ 2:50�1þ 0:0525

2

�þ 2:50�1þ 0:0525

2

�2þ 2:50�

1þ 0:0525

2

�3þ 2:50�

1þ x

2

�4þ 2:50�

1þ 0:0537

2

�5

þ 2:50�1þ 0:0537

2

�6þ 100�

1þ 0:0537

2

�6


and so it will be equal to x ¼ 5:2665%. This must be so; otherwise, arbitrage profitscould be made. If, for example, it were to yield 5.5%, all traders would buy all the sevenSTRIPs with face values of 2.50 and 100, in effect reconstituting the purchase of acoupon bond, and short the 5% coupon. This would yield a riskless profit, as thesum of the prices paid for the STRIPs would be less than 99.00.Synthetic Treasuries with any maturity and any coupon can be created through a

similar process. Suppose there were no 12% Treasury notes with 712-year maturity. A

dealer can create one by putting together a portfolio of STRIPs and coupons. The yieldon the combined synthetic security would be governed by the same arbitrage principle.Synthetic forwards can also be created by combining bought long-term Treasuries withshorted short-term ones. For example, buying a $100 million 12-year 6% coupon,shorting a $100 million 3-year 6% coupon, and buying a 3-year $100 million STRIPis equivalent to buying a security that starts paying 3% coupons 31

2years from today

and ends with the last coupon and principal 12 years from today (i.e., is a 3-year-by-12-year forward).Arbitrage is the principle behind building a Treasury curve. The Treasury zero curve

is a schedule of zero-coupon rates (or discount factors) paid by the U.S. government forany maturity between now and the maturity of the longest Treasury in existence. Theserates are used to discount cash flows and to compute a price on any other bond issuedby or collateralized by U.S. Treasuries. Any such bond can be synthesized by a portfo-lio of coupons and STRIPs. Similar to the Treasury zero curve, we can build a Treasurypar curve (i.e., a schedule of YTMs on par coupon Treasuries). The curve is used as abenchmark for quoting bonds (munis, mortgages, or corporates).Treasuries are the most common security underlying a repo contract. To the lender,

often it does not matter whether the collateral is the 2-year or the 5-year on-the-run.That is why most repos yield the same general repo yield. Sometimes, they go special,which means, because the underlying Treasuries are in demand (to own), the repolender (buying securities to resell tomorrow) will lend funds at a rate much lowerthan on other repos. That rate may even be zero. Dealers vie to own liquidity-squeezedTreasuries in order to be able to lower their cost of (borrowing) funds by repoing them.The special nature of some notes and bonds enters into the arbitrage equations whenconstructing synthetic spot or forward Treasuries.In the mid-1990s the U.S. government started issuing Treasury inflation-protected

securities (TIPS). The TIPS coupon is set to reflect a real rate of return. The principalaccrues based on the annual measure of inflation. TIPS are designed to provide aperfect purchasing power hedge. Curiously, their volume of trading is rather low.In addition to the U.S. Treasury, federally sponsored agencies issue bonds to

finance activities in a few politically preferred sectors. These government-sponsoredenterprises (GSEs) were described in the money market review. They issue uncollater-alized debentures as well as mortgage-backed securities (collateralized by mortgagepools with common characteristics). Some agencies, like Fannie Mae, in addition tobullet bonds, also issue callable bonds. Several agencies follow a calendar ofissuance (or reopening of old issues) of large bullet bonds with 2-, 5-, 10-, and 30-year maturities, similar to that of U.S. Treasury on-the-runs. These are called bench-mark bonds for Fannie Mae and the Federal Home Loan Banks or reference bonds forFreddie Mac. Today, the volume of the agency bonds traded rivals that of the U.S.Treasuries.


Government bonds in Europe and Asia

National government bond markets in Europe and Japan follow a structure similar tothat of the U.S. Primary dealers participate in auctions and then distribute bonds toinstitutional and retail investors. The auction procedures vary a little. While France andGermany follow the single-price Dutch auction based on a published calendar, theBank of England uses an ad hoc auction system, where the maturity and the totalamounts of the securities are announced only at the time of the auction. This allowsgreater borrowing flexibility and leads to less yield volatility in the markets, as old issuesare not dumped in favor of new known ones prior to an auction. Some governments usea multiple-price Dutch auction similar to that used in the U.S. prior to the 1990s wherecompetitive bids are awarded at the actual bid yield. Several governments, like theU.K., the Netherlands, and the U.S. have also used a tap system by reopening oldissues at auctions. Many governments do not issue a great variety of maturities ofbonds. In Germany and Japan, the longest benchmarks are 10-year bonds with nointervening, liquid on-the-runs. Several governments issue inflation-linked securities.In the U.K., index-linked gilts have coupons and final redemption amounts linked tothe retail price index (RPI); Canadian and Australian governments have issued bondslinked to national inflation indices. The U.K. government also issues short-term bondscalled convertibles which give the holder the right to convert to longer maturity bonds.

Government bonds around the world have easily recognizable nicknames. Amongthe major ones, U.K. bonds are called gilts and German bonds are referred to as bunds.These serve as benchmarks for pricing corporate bonds and swaps in the nationalmarkets and nearby (Swiss swaps are priced off bunds).

Secondary government bond markets around the world differ substantially in termsof their liquidity. The turnover ratio is the greatest in the U.S. where more bonds aretraded every day than the total amount outstanding. The U.S. also enjoys the smallestbid–ask spreads (less than 3 basis points). By contrast, the Swedish, Swiss, and Frenchmarkets trade with spreads of 10 or more basis points.

Figure 3.15 Size and liquidity.Data from Salomon Smith Barney; H Inouc. The structure of government securities markets in G10 countries: Summary of

questionnaire results. Market Liquidity: Research Findings and Selected Policy Implications, May, 1999, Committee on the

Global Financial System, Basel; R. McCauley and E. Remolona, Special feature: Size and liquidity of government bond

markets, BIS Quarterly Review, November, 2000. Reproduced with permission from Bank for International Settlements.


Figure 3.16 Estimated net issuance of government bonds in 2000 as a proportion of outstandingdebt.Data from Salomon Smith Barney; JP Morgan; and R. McCauley and E. Remolona, Special feature: Size and liquidity of

government bond markets, BIS Quarterly Review, November, 2000. Reproduced with permission from Bank for Inter-

national Settlements.

In any given year, the policies of central governments as to the overall new issuanceand its composition differ across countries. For example, in the early 2000s, while thegovernments of Australia, Sweden and the U.S. were paying off their debts and short-ening the maturity of their overall debt, Japan, France, and Spain were stepping up newissuance.

Corporates

Most corporate bonds are term or bullet bonds (i.e., they pay interest for a term of yearsand repay the principal at maturity). Most have final maturity dates of up to 30 years.The bond indenture may specify early redemption provisions serially, through a refund-ing, or a call. Bonds not collateralized by property, equipment, or specific revenue arecalled debentures. In case of default, they enjoy seniority over bank debt. They can alsobe subordinated (i.e., offer the least creditor protection relative to bank debt or seniorbonds). Bond information services generally classify bonds into four categories: util-ities, transportations, industrials, and financials. Prior to issue, most U.S. bonds andmore than 50% of European bonds are rated by rating agencies like S&P’s andMoody’s with the familiar letter codes starting from AAA or Aaa, respectively, allthe way down to C and D. The top four grades: prime, upper medium, medium, andlower medium, spanning S&P ratings above BB, are referred to as investment-gradebonds, the rest as noninvestment-grade, high-yield, or junk bonds (all synonyms). Ratingagencies publish credit watch lists with bonds reviewed for potential upgrades or down-grades.The origin of the modern junk bond market goes back to the 1980s which ushered in

original-issue high-yield bonds. These were rated junk from the beginning and forissuers they represented convenient longer term alternatives to private bank loans orbeing shut off from credit. To alleviate the cash flow problems of junk or leveragebuyout issuers, these bonds have often been issued with deferred or step-up coupons.


Traditionally, the largest corporate bond market has been the domestic bond sector,particularly in the U.S., where the domicile of the issuer, the currency, and the countryof issue are the same. Rapidly growing is the Eurobond sector, where investors simul-taneously come from several countries, the currency of issue is outside its homejurisdiction and bonds may be issued in unregistered form. Like domestic bonds,they are sometimes listed on an exchange, but most trade OTC. Borrowers are oftennon-financials, banks, and sovereign states. The currency of choice has traditionallybeen the U.S. dollar, but its share has steadily declined. Only some Eurobonds are Eurostraights with normal fixed coupons: many have innovative coupon structures, likezeros, deferreds, or step-ups; many are dual-currency (coupon and principal currenciesare not the same); some are convertible or exchangeable, and some have detachablestock and debt or currency warrants. Non-U.S. banks often issue dollar floating-rateEurobonds, sometimes capped or collared, and sometimes with drop-locks (i.e., auto-matic conversions to fixed).

Medium-term notes (MTNs) are corporate debt instruments, both in the U.S. andEurobond markets, offered continuously to investors by a program agent with matu-rities anywhere from 9 months to 30 years. Most MTNs are not underwritten, butdistributed on a best efforts basis. Coupon structures are very innovative, both withfixed and floating-rate features. Many MTNs are structured (i.e., they are coupled withderivatives to provide the most appealing payoff structures, like inverse floatingcoupons). With an MTN, the borrower has the flexibility to issue highly customizeddebt and can do so as needed, under a previously announced and shelf-registeredprogram. The only drawback may be the cost of registration and distribution.

Corporate bonds for different maturities trade at different rates. A corporate bondrate for a given maturity reflects the cost of borrowing/lending funds risklessly (govern-ment rate) plus an allowance for issuer default and other demand and supply issues(spread). Corporate spreads have their own term structure and are quoted relative tothe same maturity government security or swap rate. The shape of the par coupon curvetypically follows, but is not necessarily similar to the shape of the government bondcurve. Corporate spreads in the late 1990s and early 2000s reached their historicalhighs, reflecting the deteriorating credit quality and exploding leverage ratios ofprivate issuers. This has been true for all investment-grade rating categories. Duringthe same period, the spreads between the highest and lowest rating increased. Thespreads relative to benchmark swap rates also increased.

Munis

In the U.S., the primary driver behind the popularity of municipal bond (muni ) financeis the exemption of the interest received from federal income taxes. The exemptionapplies to interest income (not capital gains) on obligations of all municipalitiesbelow the federal level (i.e., states, cities, utility and highway authorities, counties,etc.). From an individual investor’s perspective, the yield on a muni can remain attrac-tive at a level substantially lower than that of a corporate or a Treasury. Muni yields arefrequently converted to taxable equivalent using the following formula:

ytax�equiv ¼ ymuni

1� �m


Figure 3.17 Corporate and government bond spreads, month-end data. Bond index yieldsagainst 10-year swap rates (in basis points), except for historical U.S. yields (in %).Data from Bloomberg, Merrill Lynch, and national data. Source: B. H. Cohen and E. M. Remolona, Overview: Are markets

looking beyond the slowdown? BIS Quarterly Review, June, 2001. Reproduced with permission from Bank for International

Settlements.

where �m is an individual’s marginal tax bracket. Municipal bonds are most popularwith high-net-worth individuals.In the U.S., the financing of schools, police, and infrastructure projects is highly

decentralized and often obtained at a local level. Governments at low levels areforced to finance their own activities, expecting little help from state or federal institu-tions. This contributes to the great proliferation of issuers. For example, Bloomberg’sdatabase lists close to 60,000 active issuers.Municipalities finance periodic funding imbalances and long-term projects using two

main kinds of bonds: general obligation (GOs) and revenue. GOs are backed by theissuer’s unlimited taxing power. Revenue bonds are backed by revenues from specificprojects (highway tolls, water authority fees).


All municipal bonds are exempt from registration requirements. Munis can be issuedas serial bonds, with pre-specified redemptions prior to final maturity, or as term bonds,but often with sinking bond provisions. Many are callable. Shorter term notes up to 3years, issued in anticipation of tax, grant, revenue, or bond fund inflows, are respec-tively labeled tax-, grant-, revenue-, and bond-anticipation notes (TANs, GANs,RANs, and BANs). Like corporates, munis are rated on their default likelihood byS&P and Moody’s using the same letter codes. It is worth noting however that manynewer hybrid bonds’ default provisions have not been tested in court and there may beno prior precedents of investors’ recovery claims.

Primary market issuance procedures parallel those of the corporate bonds. Typically,large and small issues are underwritten by large and small investment banks, orprivately placed with small groups of institutional investors. Their sales are announcedin The Bond Buyer. The secondary market is widespread but not deep. Large brokeragefirms trade in general names (nationally recognized municipal issuers), while smallerones trade in local credits, both posting their inventory offerings over the Internet.Given the variety of issuers and bond types, most munis are quoted on a yield-to-call(YTC) or yield-to-maturity (YTM) basis, rather than price. This allows comparisonsacross all bonds of the same issuer and across issuers.

Occasionally, muni yields exceed Treasury yield, despite their tax advantage. Thishappens during periods when many large issues hit the market simultaneously. Thissharply increases the supply of bonds and causes muni spreads to widen. In the summerof 2004, $15 billion of California bonds hit the market at the same time as other statesand cities issued their own bonds in order to deal with their own budget deficits.

3.3 INTEREST-RATE SWAPS

Unlike securities markets where a piece of paper originally sold by an issuer changeshands between unrelated third parties, swaps are customized private contracts witheach party exposed to the default risk of the other. Every time a swap is enteredinto, a new contract between two new counterparties is initiated. However, becauseswap conventions are so well established and because they are priced in the dealermarket to reflect the average credit quality of the counterparty (i.e., uniformly), theyare highly fungible (substitutable). Institutional counterparties willing to get out ofpreviously established swaps can unwind the transactions not only with the originaldealer, but frequently by auctioning off their side to other dealers as well. This involvesa payment or receipt of the current mark-to-market, easily established by pollingdealers, and an assignment of the swap. An unwind or assignment are not quite assimple as a purchase or sale of a security, but nowhere near as difficult as a break-up ofa non-standard private contract. The main feature stopping the swaps from enjoyingthe liquidity of government bonds is their degree of customization and legal creditissues. At the same time, their volume has increased so dramatically in recent yearsas to rival the amount outstanding of the government bonds (summarized in Figures3.18 and 3.19). This is true in terms of total swaps outstanding as well as daily cash andfutures turnover, where swap-related transactions represent nearly half of the dollarmarket.


Figure 3.18 Interest rate swaps, notional amounts outstanding (in trillions of U.S. dollars): (a)by currency; (b) by counterplay; (c) by maturity (includes forward rate agreements, whichaccount for approximately 15% of the total notional amount outstanding).Reproduced with permission from Bank for International Settlements.

Figure 3.19 U.S. treasury and other instruments in the dollar bond market, daily averagetransactions (in billions of U.S. dollars and percentages): (a) treasury futures and swap transac-tions; (b) swap and non-Treasury transactions as a percentage of bond market activity (bondmarket activity includes swap transactions. Treasury cash and futures turnover, and turnover ofU.S. dollar-denominated agencies, Eurobonds and global bonds).Data from Cedel, FRBNY, national central banks, Euroclear, FOW TRACE, ISDA, BIS estimates. Source: R. N.

McCauley, Benchmark tipping in the money and bond markets, BIS Quarterly Review, March, 2001. Reproduced with

permission from Bank for International Settlements.

When governments were paying off national debts in the early 2000s, swaps replacedgovernment bonds as de facto benchmarks for pricing other securities. This was par-ticularly true for maturities where government bond markets were thin or did not exist,like the 11-year to 25-year maturity segment of the U.S. market, and is still true in somesegments, like the Japanese market past the 10-year maturity point.An interest-rate swap is an exchange between two counterparties of streams of cash-

flows, typically resembling coupon streams of bonds. In a plain vanilla swap, one side isfixed, the other floating. The two parties agree on the final maturity of the swap (e.g., 10


years), the notional principal (e.g., $100 million), the interest rate for the fixed leg ofthe swap (e.g., 6%) to be paid on that notional principal, the floating rate index(e.g., 3-month LIBOR), the spread on top of it (e.g., þ25 basis points) to be receivedon the notional principal, and the frequency and the day-count convention of both sidesof the payments. The language of buying or selling a swap, like in securities, is neverused. Instead, one party, called a fixed rate payer or simply a payer, pays fixed andreceives floating interest amounts, while the other, called the receiver, does the opposite.The principal amount is referred to as the notional, as it can be thought of as a facevalue of two exchanged notional (i.e., fictitious) bonds. The notional in interest rateswaps is never exchanged as it is a wash (the same principal is paid and received). Nospread over the floating side is called index flat. Paying fixed on a swap is economicallyequivalent to shorting a fixed coupon bond, reimbursing the lender of the shorted bondfor missed coupons and principal, and with the proceeds buying a floating rate bondand receiving coupons and the principal on it. As the principals of both sides of theswap are the same, exchanging them at the end is not necessary. Swap settlementconvention is 2 business days, and LIBOR fixings for the floating interest paymentsare 2 days, prior to the start of the accrual period, paid at the end of the accrualperiod.

Swaps are often done on the back of bond issues. In a low interest rate period like theearly 2000s, investors tend to prefer floating rate bonds in the hope that as interest ratesgo up they can benefit. Imagine Coca-Cola issuing a $1 billion floating coupon bond,but the company really wants a fixed rate liability to focus on making syrup(!) and noton market speculation. Coca-Cola can issue a floating rate bond and enter into aninterest rate swap with a dealer like Citibank paying fixed and receiving floating,with dates of payments exactly matching the coupon dates on its bond. The floatingreceipts from the swap would then exactly offset its payments of the bond coupons,leaving Coca-Cola with a fixed-rate liability. Often, swaps are done well into a life of abond to lock in a gain or hedge against (read: speculate on) future rate movements.Suppose several years after issuing the floating rate bond, rates have actually declined.Coca-Cola can swap into fixed and pay a fixed rate lower than it would have had itswapped immediately. Many swaps are unrelated to bond issuance.

Normally, on-market swaps are entered into with no money changing hands upfront.The fixed rate (or the floating rate spread) is adjusted such that the present value of thepayments is equal to the present value of the receipts. Market-making dealers competeon the spread they quote over a benchmark for the fixed rate of the swap againstLIBOR flat. In the final negotiations, that translates into a specific fixed rate writteninto the swap’s documentation. But swaps can also be customized to match specific ratetargets. In that case, depending on whether the agreed-on fixed rate is higher or lowerthan the currently quoted swap rate for a given maturity, there may be a payment orreceipt of the present value of the swap by the dealer. Such swaps are called off-marketswaps.

Many swaps are not plain vanilla. The simplest example is when a rate paid on eitherside of the swap does not conform to its standard, like semi-rate paid annually. Otherexamples involve floating-for-floating swaps, where both sides are floating but off dif-ferent indices (LIBOR vs. FF rate), or zero-coupon swaps, where one side only accruesbut does not pay until a certain date. Chapter 4 also describes currency swaps, some-times called cross-currency swaps, where the pay side of the swap is in one currency and


the receive side in another. These can be fixed-for-floating, fixed-for-fixed, floating-for-floating or even multicurrency, referring to the types of payments and receipts.We will defer the discussion of swap valuation to Chapters 6 and 8. Here we go

through a simple example of the cash flows of a plain vanilla swap in U.S. dollars. Letus pretend that we are Citibank and, on August 19, 2003, we are entering into a 2-year$100 million swap to pay 6% semi-annually on a 30/360 basis and receive 3-monthLIBOR quarterly on an Act/360 basis. Dollar swap settlement is 2 business days, so thefixed payments will roll on May 21 and August 21, and the floating receipts will roll onNovember 21, February 21, May 21 and August 21, based on LIBOR fixings 2 businessdays prior to those dates.4 Consider Table 3.4 which is full of relevant information. Incolumns 1 and 2, we have the set dates and LIBORs we observe over the life of theswap, columns 3 and 4 contain the start and end dates for the quarterly interest accrualperiod of the floating side of the swap. For simplicity, we assume in column 5 that, ineach quarterly interest accrual periods running from August 21 to November 21 andfrom May 21 to August 21, there are exactly 93 days and, in each of the other accrualperiods, there are exactly 92 days. In real life, we would have to count the actual days.Also, if any of the LIBOR set dates were not a business day, we would have to takeLIBOR from 1 day prior or refer to the swap documentation. The semi-annual accrualperiods for the fixed leg of the swap are not shown, as each simply includes twoquarterly periods of the floating side (i.e., they run from 21 August to 21 Februaryand from 21 February to 21 August).

Table 3.4 A 2-year $100 million interest-rate swap on August 19, 2003. Pay 6% semi 30/360,receive 3-month LIBOR Act/360

Set date LIBOR Accrual period No. Floating cash flows Fixed cash flows—————————— days —————————— ——————————Start date End date Pay date Amount Pay date Amount

19-Aug-03 5.00 21-Aug-03 21-Nov-03 93 21-Nov-03 1,291,66719-Nov-03 5.10 21-Nov-03 21-Feb-04 92 21-Feb-04 1,303,333 21-Feb-04 3,000,00019-Feb-04 5.50 21-Feb-04 21-May-04 92 21-May-04 1,405,55619-May-04 5.60 21-May-04 21-Aug-04 93 21-Aug-04 1,446,667 21-Aug-04 3,000,00019-Aug-04 6.00 21-Aug-04 21-Nov-04 93 21-Nov-04 1,550,00019-Nov-04 6.20 21-Nov-04 21-Feb-05 92 21-Feb-05 1,584,444 21-Feb-05 3,000,00019-Feb-05 6.10 21-Feb-05 21-May-05 92 21-May-05 1,558,88919-May-05 5.90 21-May-05 21-Aug-05 93 21-Aug-05 1,524,167 21-Aug-05 3,000,000

Floating leg cash flows (columns 6 and 7) are based on LIBOR sets 2 days prior to thestart of the interest accrual period and are received on the end date of the accrual period(i.e., in arrears), the same way a floating rate bond would pay. Note that the firstLIBOR is known at the time of the swap; the rest, shown in italics, are not knownuntil the actual set dates. For the first interest period, we take:

100; 000; 000 � 0:0500 � 93=360 ¼ 1; 291; 667


4 Settlement day conventions vary somewhat across currencies: USD swaps use LIBOR sets from 2 business days prior, whileGBP swaps use same day sterling LIBOR sets and the effective date is the trade date.

Fixed rate leg cash flows are all based on the same quoted 6% rate and are paid on theend dates of the semi-annual interest period, the same way a fixed coupon bond wouldpay. For each period, we take:

100; 000; 000 � 0:0600 � 180=360 ¼ 3; 000; 000

For an on-market swap, the present value of the fixed cash flows is equal to thepresent value of the floating cash flows, where we substitute today’s forward 3-monthrates for the unknown future LIBOR sets. We explain the logic of that arbitragecondition in Chapters 6 and 8.

Like bonds, swaps for different maturities trade at different rates and reflect the costof borrowing/lending funds risklessly (government rate) plus an allowance for counter-party default and other transaction-related issues (spread). Swap spreads have theirown term structure and are quoted relative to the same maturity government security.The shape of the swap (par coupon) curve typically follows, but is not necessarilysimilar to, the shape of the government bond curve. Swap spreads in the late 1990sand early 2000s reached their historical highs, reflecting the deteriorating credit qualityof an average counterparty.

Figure 3.20 Yield curves for interest rate swaps (in percentages): (a) U.S. dollar; (b) euro;(c) yen.Data from Bloomberg. Source: B. H. Cohen and E. M. Remolona, Overview: Are markets looking beyond the slowdown?

BIS Quarterly Review, June, 2001.

3.4 MORTGAGE SECURITIES

Mortgage-backed securities (MBSs) are obligations of an issuing agency or a financialinstitution fully collateralized by the stream of interest and principal payments of theunderlying mortgage pool. Traditional mortgages in the U.S. have been fixed interestfully amortized loans with level monthly payments. These give borrowers the right andincentives to prepay the loan prior to its maturity which is typically 30 years (less often15). To understand the mechanics and risks of this segment of fixed income markets, itis useful to review the basics of real estate financing.

A purchaser of a house, after supplying a down payment from his own funds (typic-


ally 20% but often less) obtains a significant portion of the funds (the loan-to-valueratio) needed from the original lender of funds, the mortgage originator. The originatormakes money by charging an origination fee and other fees and by selling the mortgagein the secondary market to a larger financial institution for a higher price. This allowsthe originator to replenish his funds to make new loans. The new owner of the mortgageloan may outsource the collection of the monthly payments to the originator or a thirdparty by paying a servicing fee. The mortgage investor buys many similar loans. He hasan incentive to further sell the collected pool of mortgages in order to be able to acquirenew loans (otherwise, he would quickly run out of funds to invest). He assembles poolsof mortgages according to government agency guidelines and sells them to agencies forthe purposes of securitization or issues collateralized securities directly to investors. Theonly spoiler in this process can be the homeowner himself who has the right to prepaythe mortgage partially (by sending more money each month) or completely (e.g., byselling the property). He does so especially when interest rates drop and he can re-finance at a lower rate. But the homeowner is not an efficient refinancer and he prepaysfor unique reasons too (e.g., inheritance); in a pool of mortgages there will be some thatpay off fast, some that pay off slowly. In general, the prepayment speed will increase asinterest rates go down. The prepayment speed is measured by a standard unit called aPSA (public securities association): 100% PSA means that a pool prepays at a rate of6% per year if the mortgages are over 30 months old or by a percentage, starting at 0%all the way to 6%, increasing by 0.2% for the first 30 months, that percentage reflectingthe fact that newly issued mortgages rarely prepay.The securitization of mortgages takes two basic forms. The first stage is to issue pass-

through certificates. Pools of similar mortgages (by geography, size, type of dwelling,etc.) are placed in a trust entity and certificates entitling holders to a proportional shareof monthly receipts are sold (98% of pass-throughs are created by the agencies, the restby financial institutions). As homeowners prepay their loans, particularly in a lowinterest rate environment, the effective duration of mortgages shrinks with investorsreceiving larger amounts in the early years of the pass-through and next to nothingtoward the end. The greatest risk of a mortgage security is this implicit call option soldto homeowners. This entails contraction risk if prepayments speed up or extension riskif they slow down relative to baseline PSA. The default risk is minimal as all underlyingloans are mortgages with full recourse rights of the lender to foreclose and sell theproperty.In order to create securities that enjoy the default protection of mortgages but are

relatively immune to prepayment risks, the second stage of securitization involves theissuance of collateralized mortgage obligations (CMOs). Pass-throughs are put in atrust, and several distinct tranches of bonds with fixed coupons are issued. The sumof the face values of all the tranches is equal to the face value of the collateralpass-through. However, monthly payments received from the collateral are sliced andallocated to the tranches in the order of pre-assigned priority. For example, in asequential-pay CMO, all interest payments are allocated equally to all tranches basedon the assigned principal, but all the principal payments go first to paying off tranche A,then B, etc. Tranches C and D may not receive principal payments until later years. Theeffective duration of the tranches depends greatly on the speed of prepayments. Earlytranches enjoy extension risk protection. Later tranches enjoy prepayment risk protec-tion, but also face the most uncertainty as to the timing of principal retirement.


A common variation on the sequential pay CMO structure is one with an addedresidual accrual tranche called a Z bond. The accrual tranche initially does not receivecurrent coupon interest. Instead, interest is accrued and added to the principal to be paidwhen the tranche starts receiving distributions. Another variation ensuring a greaterstability of the average life of a bond is a CMO that includes a planned amortizationclass (PAC) bond as the first tranche and the support of companion classes as the othertranches. In this CMO, payments are diverted from the companion classes to ensure thatthe PAC class follows a pre-specified principal repayment schedule based on a statedPSA percentage. This protects against both contraction and extension risk of the PACclass, with most of the risk shifted to the companion classes.

MBSs also trade in stripped forms, most often as interest-only (IOs) and principal-only (POs). Similarly to the sequential and PAC CMOs, pass-throughs are put in a trustthat issues certificates entitling holders to the portions of the monthly payments separ-ately relating to interest and principal. Thus an IO holder receives the coupon interestportion only, calculated on an ever-declining principal, and, if prepayments speed up,the total sum of receipts may be lower than the original amount paid for the IO. On theother hand, if prepayments are slow, the IO holder may receive payments for the fullmaturity of the mortgages. The unique feature of an IO is that its value rises as interestrates rise, which slows the prepayments. POs are sold at a discount from face value andbenefit from interest rate declines that speed up prepayments. Their values are highlyinterest rate-sensitive.

The valuation of MBSs is a mix of PV calculations and statistics. The statistics partrelates to prepayment modeling, which involves estimation from past history of home-owner responses to financial and non-financial variables. As we mentioned before, thehomeowner has the right to prepay equivalent to the right to call the loan at par (like anissuer of a callable bond). In most models, it is assumed that the homeowner comparesthe rate on his mortgage to the 7- or 10-year rate on U.S. Treasury, which is thebenchmark for 30-year fixed rate mortgages. However, more often than not the home-owner does not exercise this option as soon as it is optimal to do so. He may have justrefinanced, refinancing may be costly, or the homeowner may have other family issues.Sometimes, the homeowner exercises early (e.g., when his family relocates and theproperty is sold). Even if the mortgagee is interest rate-sensitive and tends to exerciseas rates decline, he may take other factors into account, like the last date of refinance,the rate change not the level, rates on alternative mortgage structures, etc. Homeownersalso vary in some individual characteristics, like age, income, location, type of property,etc. Even when a statistical model of prepayments is constructed, the valuation of anMBS typically involves more than present value discounting of cash flows based onestimated principal. Because the principal itself is not interest rate-level but path-dependent, the valuation involves a Monte Carlo or tree simulation, or some othervaluation technique similar to option pricing. The excess yield spread over a staticpresent valuing is referred to as an option-adjusted spread of an MBS. It is at best anestimate value of the implicit options held by the homeowner.

3.5 ASSET-BACKED SECURITIES

Similar to MBSs, asset-backed securities (ABSs) are securities collateralized by receiptsfrom other underlying assets. The most popular are home equity loans, student loans,


auto loans, credit card debt, and, lately, other bonds, like high-yield, distressed debt,and emerging market bonds. The underlying securities or loans, which can be amortiz-ing or not, are placed in a trust or a special purpose vehicle (SPV). Often the SPV isover-collateralized as a form of internal credit enhancement. Interest rate paid on ABSscan be fixed or floating, reflecting the nature of collateral. The structure can be pay-through, analogous to MBS pass-throughs, where each ABS represents a fractionalinterest in the cash flows of the collateral, or tranched with one senior and severalsubordinated tranches. Almost all ABSs are subject to prepayment risks. Home-equity ABSs have published prospectus prepayment curves (PPCs) similar to PSAschedules. Auto loan ABSs are measured in terms of absolute prepayment speed(ABS). Auto loan-backed ABSs are issued by subsidiaries of auto manufacturers,banks, and finance companies. Many student loan ABSs enjoy at least fractionalgovernment default protection. Credit card receivable-backed ABSs, sold by banks,retailers, and travel companies, are structured as master trusts of principal andfinance charge receivables. The issuer sells several series from the same trust, oftenby randomly selecting accounts. Credit card ABSs are non-amortizing due to a lock-out period provision. During the lock-out the principal repayments are reinvested inadditional receivables.The U.S. has the largest ABS market, but the last 5 years have seen an explosive

growth of the market in Europe. A new but growing segment is the corporate asset-backed market, originating mainly in the U.K. The largest asset used for backing,reflecting its large market value, is commercial real estate, accounting for 32% of thetotal. But the market enjoys a wide variety of assets used for backing, includingequipment and auto leases, aircraft leases, trade receivables, and others.Short-term ABS spreads have historically exceeded long-term spreads, reflecting the

expected relative price stability of the longer term assets backing the bonds. Whileunsecured borrowing spreads for European corporates have increased and becomeincreasingly variable, due to the credit problems of the issuers, ABS spread averageshave remained relatively stable.

Figure 3.21 The increasing prominence of corporate-related securitization in Europe.Reproduced with permission from Ganesh Rajendra (Managing Director, Global Markets Research, Deutsche Bank AG)

and White Page, author and publisher, respectively, of The Use of Securitization as an Alternative Funding Tool for

European Corporates.


Figure 3.23 Spreads in the unsecured bond market and ABS market compared.Rproduced with permission from Ganesh Rajendra (Managing Director, Global Markets Research, Deutsche Bank AG) and

White Page, author and publisher, respectively, of The Use of Securitization as an Alternative Funding Tool for European

Corporates.

One category of ABS that has emerged only in the last 10 years is the collateralized debtobligation (CDO), known in its two guises: a collateralized bond obligation (CBO) andcollateralized loan obligation (CLO). A CBO is backed by a diversified pool of high-yield or emerging market bonds; a CLO is backed by a pool of bank loans or distresseddebt. An asset manager sponsoring the CDO for arbitrage or balance sheet reasons


Figure 3.22 European corporate securitization: (a) by country (1999–2001 cumulative); (b) byasset type (1999–2001 cumulative).Reproduced with permission from Ganesh Rajendra (Managing Director, Global Markets Research, Deutsche Bank AG)

and White Page, author and publisher, respectively, of The Use of Securitization as an Alternative Funding Tool for European

Corporates.

typically sells three tranches of certificates in the order of cash flow claim: senior,mezzanine, and subordinate equity. The funds obtained from the sale of certificatesare used during the ramp-up period to acquire collateral. The division allows the seniortranche to obtain a rating that is higher than the average of the underlying collateral.The pool is managed to generate the requisite cash flows during the life of the CDO.


___________________________________________________________________________________________________________________________________________________________________________ 4 __________________________________________________________________________________________________________________________________________________________________________

________________________ Equities, Currencies, and Commodities ________________________

Spot markets in securities other than fixed income are simpler and more complex at thesame time. The underlying mathematics are much simpler because these securities donot normally involve multiple cash flows (a currency does not pay coupons like a bond)and there are few or no close substitutes that can be repackaged to create a stock, acurrency, or a commodity (like coupon bonds with zeros). Each spot currency or stockis unique and its value cannot be mathematically related to that of another currency orstock. There is no stated maturity value, and thus there is much greater uncertaintyabout the future redemption price. Because they rely on estimates of future economicvariables, cash flow-discounting techniques for valuing stocks or commodities are atbest good approximations of the fundamental value. Current market prices are notstrict functions of those cash flows the way bond prices are of coupon streams. Atbest, they are educated guesses based on economic analyses of supply and demand,growth assumptions, and a lot of other subjective measures. The upshot is that thediscounting techniques are relatively simple, but imprecise, as they rely on a lot ofuncertain economic variables.

4.1 EQUITY MARKETS

Secondary stock trading around the world is arranged to force buyers and sellers ofeach stock into one marketplace in order to discover the best price for the stock. Thebottleneck design ensures that all relevant information about the stocks reflected in thebid and offered prices flows into one place. This helps establish the price that reflects thebalance of all demand and supply for the stock. It also guarantees maximum liquidityof trading, allowing traders and investors to buy and sell nearly instantaneously andwith minimal transaction costs. The bottleneck can be designed in two ways: one isthrough a physical exchange; the other is through a virtual exchange which is really anetwork of dealers linked by phones and computers. The first typically creates a greaterconcentration of supply and demand, while the second offers more competition.Primary equity sale works roughly the same around the world. New issuers of stock

contact investment bankers who help them navigate through the regulatory process andhelp them sell the stock certificates to investors. In most countries new shares have to beregistered with a national government regulator to ensure a minimal level of informa-tion disclosure. The stock is then offered to investors as a fully underwritten issue or ona best efforts basis. In the former, the investment banking firm buys the entire issue andthen sells it to investors out of its own inventory. In the latter, the investment-bankingfirm does not take all the resale risk. Instead, it only sells as much of the issue as it can.

The process is the same whether the issues are offered in an initial public offering (IPO),where new companies sell stock for the first time, or in a seasoned equity offering(SEO), where well-established firms distribute additional shares to new investors. Theflip side of the primary market is the repurchase of shares by the original issuers, or abuyback. This is done in the secondary market, but is normally preceded by anannouncement of a buyback program. Buybacks permanently reduce the number ofshares circulated among investors.

Compared with markets for bonds, commodities, or derivatives, equity markets arewell understood by novice investors. We focus on some interesting recent developments.

Secondary markets for individual equities in the U.S.

The largest and most important secondary market for individual equities is the NewYork Stock Exchange (NYSE). The NYSE processes about 1.4 billion shares every dayand derives its strength from a unique setup based on specialist trading posts. The floorof the NYSE consists of trading booths, each trading in a handful of stocks assigned tothem. In order to provide the maximum liquidity of trading (force buyers and sellers tomeet at one bottleneck spot), each stock is assigned to only one booth and one author-ized dealer for that stock—the specialist sitting in the booth. Most orders arrivethrough and are electronically matched by a SuperDot computer system. The restcome from the crowd of brokers gathered around each post. Unmatched limit andstop orders are entered into the limit order book maintained by the specialist. Theyare prioritized based on their size and best execution price. The specialist is in a uniqueposition to see who wants to buy and sell and at what prices. Since 2002, the crowd hasbeen allowed to see the total size and price for each bid and offer. The specialist enjoys amonopoly power to trade for his own account by observing the flow of trading in thelimit order book. For the privilege of being able to profit from knowing the flow of buyand sell orders for his stock, the monopolist is obligated to ‘‘maintain fair and orderlymarket’’ (i.e., sometimes lose money when trading imbalances occur). He is a marketmaker and must post bid and offer prices at all times. He is forced to trade when he maynot want to. This ensures the continuity of the price and reasonable depth of quotes. Itmay seem unfair to allow the specialist to profit at the expense of public investors, andrecent scandals over the abuse of monopoly power by specialists may force the NYSEto re-examine its reluctance to adopt a dealer model. The specialist’s monopoly profit isthe price for continuous price discovery. The Tokyo Stock Exchange is the best exampleof where machines sometimes fail to deliver. All buyers and sellers are forced to submitorders into the same limit order book. Market participants have electronic access to it(they can see the size and the identity of the bidders and sellers), but if at any given timethere are no market orders and no seller wishes to trade at a price equal to the highestbid, then no trading occurs. At the NYSE, the specialist must step in to trade.

There are disadvantages to this arrangement. With the advent of decimal trading(quoting prices in dollars and cents as opposed to eighths and sixteenths of dollars) andmore importantly the reduction of the tick size (minimum price increment) to 1 cent,specialists have had incentives to front-run investors by showing prices only 1 centbetter than the best bid or offer. Front-running tips the balance of fairness awayfrom the investor and in favor of the specialist. The NYSE has had to police the


specialists more to ensure that the tradeoff of monopoly profits for continuous pricemaking is balanced.NYSE, also called the ‘‘Big Board’’, lists over 2,000 U.S. and foreign stocks. The

American Stock Exchange, the AMEX or the ‘‘Curb’’, lists about 700 mostly smallercompanies (in terms of market cap), but has become known for its innovation inwarrants and a variety of standardized stock baskets. Both the NYSE and theAMEX are physical exchanges based on a single market-maker auction. In contrast,the U.S. over-the-counter (OTC) market called NASDAQ, owned and operated by theNational Association of Securities Dealers (NASD), has no single physical location andis based on multiple market-makers who quote and trade shares through a computer-ized automatic quotation system (�AQ). There are well over 4,000 stocks listed on thetwo tiers of the NASDAQ (national and small-cap). Just like the physical exchanges,the NASD maintains listing requirements (albeit less stringent and costly), yet theNASDAQ stocks are referred to as ‘‘unlisted’’. All NASDAQ members are dealers(i.e., they trade for their own account, and many trade the same stocks). They areobligated to continuously display firm bids and offers good for 1,000 shares and,when hit, trade at those quotes with other members. The practice common onNASDAQ, but not permitted on the exchanges, is the internalization of many customertrades. The dealer may execute customer orders acting as a principal and not show theorder to other dealers. This takes liquidity away from the central market which isalready, by design, fragmented. The positive of this setup over the single auctionspecialist is the competition multiple dealers provide. This may not be enough tocompensate for the shortcomings of diminished liquidity as evidenced in NASDAQ’swider bid–ask spreads relative to NYSE.The vast majority of OTC issues (about 8,000) not listed on NYSE, AMEX, or

NASDAQ trade on the OTC Bulletin Board (most) or on the electronic version ofold ‘‘Pink Sheets’’ (penny stocks). Both of these are reported by the NASD. In addition,in the so-called ‘‘third market’’, exchange-listed stocks trade OTC via a system similarin functionality to and operated by the NASD.Since the 1990s, a fourth market has emerged in the form of for-profit brokers’

networks. To avoid exchange commissions, customers can meet directly in fully elec-tronic markets maintained by third parties. These can be either electronic communica-tions networks (ECNs) or crossing networks. ECNs operate within the NASDAQsystem (representing 30% of volume) as widely disseminated electronic limit orderbooks. Subscribers link to them to anonymously post quotes and execute trades. Thelargest ECNs include Instinet, Island, and Archipelago. ECNs are economical ways tohandle small retail orders. Crossing networks are designed for institutional investors.They match buyers and sellers directly by aggregating orders for execution as a batch ata specified time. They are popular with mutual funds.

Secondary markets for individual equities in Europe and Asia

Lacking a single, integrated market for equity capital, European companies list onnational exchanges or cross-list on several European exchanges. The London StockExchange (LSE) has struggled to maintain its lead in having the most internationalequities traded on its floor (about 20% of issues). The trading mechanism includes bothan order book and multiple market makers. Dealers are linked through a screen

Equities, Currencies, and Commodities 103

quotation system, but matched bids and offers in FTSE 100 shares are executed auto-matically. Any appropriately capitalized firm can compete as a dealer in multiple stockson the LSE. The order-matching book of the LSE is known as the SETS and the dealerquotation system as the SEAQ. In Germany, stock trading is concentrated on theDeutsche Borse which combines floor and electronic transacting. Buyers and sellersmust submit their trades through a bank represented on the Borse. Stock trades areconsidered banking transactions according to the German universal bank tradition.The exchange operates three segments: the official market, the regulated market, andthe Neuer Markt for smaller companies. Xetra is the electronic order-matching systemoperated as the central limit order book. All European countries operate nationalexchanges. Recently, competition among them has forced many to seek alliances aseach tries to become the continent’s dominant one. Euronext combines the Amsterdam,Brussels, Lisbon, and Paris exchanges and is second to LSE in the number of inter-national stocks listed. Spain’s exchange lists most companies in Europe, mostly domes-tic. Not a single European exchange is based on the specialist posts. Only Amsterdamaffords that status to some dealers (Montreal and Toronto also have single-auctionmarket-makers). Most are based on a competitive dealer principle. This means, forexample, that London dealers may take larger positions in stocks than dealers in NewYork. This is not so in Paris because the automated trade systems do not afford enoughanonymity to block trading firms. Clearing of trades in Europe is fragmented alongnational lines and contributes to the high cost of executing transactions.

The Japanese stock market, over 120 years old, consists of six exchanges, the domi-nant one being the Tokyo Stock Exchange (TSE). Over 1,800 domestic TSE shares areclassified, based primarily on size, into two sections, the majority being traded in thefirst section. In the continuous-auction TSE market, the ‘‘Saitori’’ intermediaries, whoare not allowed to trade for their own account, maintain the computerized centralbook. They match orders based on price priority and time precedence and rely on anopen outcry to solicit new orders. All securities must be traded by an authorized dealer.The three major dealing firms are Nomura, Nikko, and Daiwa.

In China, the principal exchanges exist in Shanghai and Shenzen. They trade class Ashares denominated in the nonconvertible yuan and class B shares denominated in U.S.dollars and HK dollars, respectively. In 2001, the government permitted the ownershipof B shares by Chinese nationals via hard currency accounts and allowed state-ownedcompanies to sell B shares. However, property rights and accounting rules havecontinued to hamper trading in B shares. In addition, companies are tied strongly tothe Chinese economy trade on the Hong Kong exchange. They are often referred to asH shares.

Depositary receipts and cross-listing

As of 2001, there were more than 1,800 sponsored depositary receipts (DRs) fromalmost 80 countries1 traded around the world. Depositary receipts are claims toshares of foreign companies. In the U.S., American depositary receipts (ADRs) arecreated by a bank, like the Bank of New York, by accumulating shares of foreigncompanies. The claims on those shares, one for one, are then traded on the stock


1 See the Bank of New York’s site at www.adrbny.com

exchanges as if they were shares of domestic stocks. They are convenient for investors;they trade in U.S. dollars, their dividends are paid in dollars, and they can be boughtand sold in the U.S. market lowering the legal risk and cost. DRs can be sponsored by aforeign company actively seeking listing and selecting one authorized depositary bankor non-sponsored (i.e., created by any bank). Legally, a DR holder may give up votingand pre-emptive rights. Creation of DRs allows companies around the world to cross-list on several national exchanges in order to tap new sources of capital. For example,large Swiss or Dutch multinationals choose to have their shares traded simultaneouslyon their small domestic exchanges, as well as in London and in New York.

Stock market trading mechanics

In most markets around the world, investors can enter several types of orders. The mostimportant are market orders and limit orders. Market orders are orders to buy at thecurrent offer price and to sell at the current bid price a given number of shares of stock.The number of shares is normally specified in terms of whole lots (normally 100 or1,000 shares). The customer specifies just the ticker symbol of the security and thequantity to be bought or sold. The price at which he will buy/sell is that currentlyquoted when the order arrives on the floor of the exchange or is posted on an ECN.The market order is executed immediately against existing limit and market orders.Limit orders are orders to buy or sell a specified quantity of a stock at a price equal to orbetter than a specified limit. Thus, in a limit order, the customer must specify the priceat which he is willing to trade.For example, a limit order may read ‘‘Buy 200 shares of IBM at 76.75.’’ If there are

other customers’ orders in the book to buy IBM at 77, they will be executed ahead ofthe limit order at 76.75 as they enjoy a price priority. If there are no better bids and thelowest offer is to sell at 77.25, then the limit order at 76.75 will not be executed until amarket sell order or another limit sell order at 76.75 or lower are entered into the orderbook.All orders can have additional instructions like ‘‘good for the day’’, ‘‘good till

canceled’’, all or nothing, etc. In most markets, orders are entered into a centralbook and prioritized based on price. The highest bid is waiting to be executed first,then the second highest bid, etc. The lowest offer is waiting to be executed first, then thesecond lowest offer, etc. The time of the order arrival has only secondary precedence(i.e., only for orders at the same price). This priority arrangement contributes toachieving the smallest spread between the lowest offer and the highest bid. The sizeof the spread is a measure of market liquidity, and the cost of trading is as real as anyexplicit commission, fee, or duty. For large institutions, anonymity is of paramountimportance in minimizing impact costs. A large block for sale by a well-known dealermay cause the selling price to drop and thus lead to lower total proceeds. The upstairsmarket in New York attempts to alleviate the impact of block trades that constituteabout 50% of all volume on the NYSE. There, block traders (minimum 10,000 shares)arrange transactions directly with each other, with only the unmatched balance exe-cuted through the floor crowd.There are also explicit costs of trading stocks, like commissions and clearing fees.

Often these cover ‘‘free’’ services customers get from brokers, like access to researchreports and trade data. These are commonly referred to as ‘‘soft dollar’’ payments as


they are hidden in trading fees. Regulators, like the Securities and Exchange Commis-sion (SEC) in the U.S. or the Financial Services Authority (FSA) in the U.K., limit thetypes of implicit perquisites investors can obtain in this manner. In the OTC markets,market makers offer cash payments to brokerages with access to customer trades. Thisentices brokerages to send all their orders to a select few market makers in order toaccumulate commission rebates. This practice is commonly referred to as payment fororder flow. It continues to be controversial as it provides perverse incentives for brokersto send orders to markets with the highest level of these ‘‘kick-backs’’ as opposed tothose with the best execution price. It can be argued that they contribute to unjustifiablyhigh bid–ask spreads. It can also be argued that they represent a fair payment for accessto information flow. It is doubtful that cost benefits are passed on to retail customers.

Stock indexes

Investors around the globe follow stock markets with the help of stock indexes. Today,some of them are household names: Dow Jones, S&P 500, and NASDAQ in the U.S.,FTSE 100 in the U.K., CAC-40 in France, DAX in Germany, Nikkei 225 in Japan,Hang Seng in Hong Kong, All Ords in Australia, to name a few. Stock indexes reflectthe price movement of diversified baskets of stocks in each national market. Most arecapitalization weighted (notable exceptions are the Dow Jones in the U.S. and theNikkei 225 in Japan which are price-weighted) to ensure proper weighting (by size)of all companies in the basket. Sometimes the baskets are not as diversified as mostbelieve. The Dow Jones and DAX reflect only 30 stocks. The Nikkei has 225 stocks.The CAC-40 has only 40. In most markets, better alternatives exist, but are lesspopular. In the U.S., the Wilshire 5000, Russell 1000 and 2000 indexes are broaderindexes. In Japan, TOPIX includes all first-section TSE stocks. There is also a variety ofindex indicators covering major world regions. The most widely used of those are theMorgan Stanley Capital International and Financial Times World Indexes that dividethe world into logical economic regions roughly based on continental boundaries.These are calculated in local as well as major currencies.

The publishing of indexes is a business run by private companies or exchanges. Oftencompanies vie to be included in some indexes in a belief that doing so increases theliquidity of their stocks. Index companies charge fees to bond and warrant issuers andderivatives dealers for referencing the index names in settlement documents of con-tracts. Indexes are a basis for many futures and options contracts, both exchange andOTC-traded. There have been hundreds of principal-protected bond issues whosecoupons are linked to stock indexes that have been sold to retail investors in Europeand the U.S. in the last 10 years. There are hundreds of mutual funds that compete onpassively replicating the indexes at the lowest cost.

A capitalization-weighted, or cap-weighted, index is created by computing a weightedaverage price of a basket of stocks with all stocks included in the index taken inproportion to their total market value (stock price� the number of shares outstanding),initially divided by some constant number to normalize the index. Every day as theprice of each stock changes the index value will change as a result of the price changeitself and indirectly of the total market capitalization change. Funds, attempting toreplicate indexes, continuously monitor the proportions of their holdings, even thoughthey are largely self-adjusting. It is important to note that even though the S&P 500 is a


cap-weighted index of the 500 largest U.S. companies, their inclusion in the index is notautomatic: rather, it is decided by a selection committee run by Standard and Poor’s.

Exchange-traded funds (ETFs)

Until a few years ago, small investors hoping to replicate the performance of a stockindex could only buy index mutual funds. The largest one of these in the U.S. is theVanguard Equity Index 500 (ticker: VFINX). The fund itself could be bought and solddirectly from Vanguard at the end-of-the-day net asset value (NAV). The NAV of afund is calculated by taking the total value of all the holdings using the market prices atthe close of a business day (4 p.m. New York time) and dividing it by the number ofshares issued by the fund. The portfolio manager for VFINX would try to replicate theperformance of the S&P 500 index, charging only minimal running costs subtractedfrom the value of the holdings in the calculation of the NAV. Recently, funds similar inobjective to index mutual funds started trading directly on the exchanges. The AMEXlists over 100 such funds. These ETFs can be bought and sold like ordinary stocksthroughout the day in the secondary market (investor-to-investor). Each share repre-sents a mini closed-end (trust) fund representing a claim to a basket of stocks resem-bling the targeted index. The funds are run by depositary banks (the State Street Banksponsors S&P 500 Depositary Receipts, or SPDRs) or asset management companies(Barclays Capital sponsors iShares), including those managing traditional open-endmutual funds (Vanguard). The institutions charge stated management fees reflectedin the current prices of the funds. The most popular ETFs are ‘‘Cubes’’ (ticker:QQQ), ‘‘Spyders’’ or SPDRs (SPY), and ‘‘Diamonds’’ (DIA) replicating the perform-ance of NASDAQ 100, S&P 500, and the Dow, respectively. There are other indexETFs, like iShares-Russell 2000 (IWM), and foreign stock and sector ETFs, likeiShares-MSCI Japan (EWJ) or Select Sector SPDR-Financial (XLF).In Europe, after an initial run by investment banks to create a variety of

ETFs, the industry underwent a consolidation, reflecting the low-cost nature of thebusiness.

Custom baskets

The most widely popular custom baskets in the U.S. are HOLDRS (pronounced‘‘Holders’’, short for holding company depositary receipts) traded on the AMEX. AHOLDR is a depositary receipt issued by a trust representing ownership interests in thecommon stock or ADRs of companies involved in a particular industry, sector, orgroup. HOLDRs allow investors to own a diversified group of stocks in one investment(like mutual funds or ETFs). The composition of a HOLDR is specified in a prospectusand typically consists of 20–50 stocks. As of May 2003, the following HOLDRs weretraded on the AMEX:


Table 4.1 AMEX-listed HOLDRs

Name Symbol

B2B Internet BHHBiotech BBHBroadband BDHEurope 2001 EKHInternet architecture IAHInternet HHHInternet infrastructure IIHMarket 2000þ MKHOil service OIHPharmaceutical PPHRegional bank RKHRetail RTHSemiconductor SMHSoftware SWHTelecom TTHUtilities UTHWireless WMH

Source: The AMEX website: www.amex.com/holdrs/prodInf

HOLDRs can be disassembled by their owners. For a small fee, investors can turn themin and obtain individual shares underlying each ‘‘mini-trust’’.

In addition to exchange-traded baskets designed for retail investors, brokers executebuy and sell orders from large investment companies on very customized, one-offbaskets. Institutions transmit the composition of the basket to be traded. The basketmay contain stocks from different exchanges. Brokers then execute program trades (buyor sell orders for a group of at least 15 stocks with a total market value of at least $1million, sent electronically to the SuperDot computer of the NYSE) or ‘‘work theorders’’ according to customer instructions (e.g., trade the order throughout the dayon the NASDAQ in an attempt to get the best price or to find buyers). The competitionis based on speed and execution costs which often run at less than 3 cents per share inthe U.S. Program trading is also used for an index arbitrage strategy involving indexfutures contracts (see Chapter 7).

The role of secondary equity markets in the economy

One of the main reasons regulators around the globe endeavor to establish and main-tain orderly stock exchanges is their role in facilitating the flow of capital from savers tonew productive ventures (businesses), even though exchanges do not participate in thatprocess directly. Most stock market regulation is aimed at investor protection whichincludes financial disclosure, fraud prevention, and protection of the property rights ofsmall investors. Exchanges are ‘‘used merchandise’’ markets in that capital flows frominvestor to investor. None of it goes to companies that issue shares to finance manu-facturing or service businesses. That initial transfer of capital takes place in the IPO orSEO market. There, newly issued shares are sold to investors. Stock exchangesindirectly play three paramount roles in this process. First, the original IPO or SEOinvestors know that they will not be stuck with their investments for ever. They can


share their risk by selling their stakes to others. By going to a broad market with manybuyers and sellers, they are reasonably assured to get a fair price for their shares.Second, the exchanges provide fair price discovery not only for investors, but alsofor the issuers. Companies often contemplate share buybacks or seasoned offeringsas ways of shrinking or expanding the equity capital invested in their businesses. Thesecondary equity markets provide a way of estimating the cost of that capital inthe same way that bond markets offer a way of discovering the interest rate on thecompany’s debt. The third role for exchanges is as a risk-sharing mechanism for venturecapital firms who use IPOs as their exit strategies. Their incentives to invest would bemuch reduced if the only exit available were to find other private investors.

4.2 CURRENCY MARKETS

The currency or foreign exchange (FX) markets are borderless. Spot and forwardstrading takes place around the globe 24 hours a day. FX trades OTC (i.e., via aphone and computer network of dealers scattered in various locations around theworld). As of 2001, the global turnover amounted to over $1 billion per day, ofwhich 33% occurred in the U.K., 17% in the U.S., 10% in Japan, 7% in Singapore,6% in Germany, 5% in Switzerland, and most of the rest in the developed nations ofEurope and Australasia. About one-third of the daily turnover is spot trading, the restis short-term currency swaps and outright forwards. During the day, trading volumeshifts slowly westward through three zones: Australasia, Europe, and North America.The market has two segments: the wholesale or interbank market, dominated by over100 international banks and some of the largest investment banks, and the retail marketin which international banks trade with their customers. Two vendors, Reuters andEBS, are leaders in currency quote systems for the wholesale market. Most of dailyvolume of trading in the interbank market can be attributed to speculation andarbitrage. Inventory adjustments and central bank interventions represent only asmall part. Clearing takes place via a private Brussels-based SWIFT (Society forWorldwide Interbank Financial Telecommunications) system or a combination of theCHIPS (Clearing House Interbank Payments System) network and Fedwire for U.S.dollar payments. Spot settlement in FX markets is 2 days hence, except for USD/CADwhich is 1 day.The U.S. dollar continues to be the dominant currency in the world. As of 2001, 90%

of all FX transactions involved the U.S. dollar on one side, 38% involved the euro, and23% the Japanese yen. Most currencies are quoted in European terms (i.e., in foreigncurrency per dollar). British Commonwealth currencies and the euro are quoted inAmerican terms (i.e., in dollars per currency). With some exceptions, currencies arequoted to four decimal places except for larger numbers which are quoted to two. Thefirst widely known significant digits, called the big figure, are often skipped. Only thesmall figure or points are quoted. For example, an FX rate of [USD/GBP] 1.4556,would be quoted on the phone as 56, with both parties knowing the implicit 1.45.The standard quote size for major hard currencies in the interbank market is $10million referred to as ‘‘10 dollars’’.Currency quotes are prices. Normally, the price of a good is expressed in terms of

currency per unit of the good. Potatoes may be priced in dollars per pound or euros per


kilo. Rarely does one see potato prices expressed in terms of how many kilos 1 eurobuys. With currencies, the pricing good and the priced good are both currencies, andthe price can be expressed both ways. In Chapter 2, we introduced the notation of spotquotes as X ’s followed by the quotation terms in square brackets (e.g., euros per U.S.dollar as [EUR/USD]) or as superscripts. Currencies can be easily converted fromAmerican terms into European terms and vice versa by taking reciprocals. Forexample, if one day the British pound (GBP) is quoted as: X ½USD=GBP� ¼XUSD=GBP ¼ 1:5000, which means that one pound buys 1.50 dollars, then that neces-sarily means that X ½GBP=USD� ¼ XGBP=USD ¼ 1=1:50 ¼ 0:6667 (i.e., 1 dollar buys0.6667 pounds sterling). In general:

X ½CURR1=CURR2� ¼ 1=X½CURR2=CURR1�The conversions are easy, but can be confusing. Suppose X ½CHF=USD� ¼ 1:7200. Howmany dollars will 300 francs buy? Since we want the quote in dollars per franc, we needto invert the quote and multiply by the number of francs:

300 � X ½USD=CHF � ¼ 300 � 1

X ½CHF=USD� ¼ 300 � 1

1:7200¼ 174:42

300 francs buys 174.42 dollars.Like any market, FX trading is subject to a bid–ask spread. We are all familiar with

the most egregious example of that from the airport currency counters. Suppose for theUSD/GBP you observe: 1.4556/1.4782. But, which is the bid and which the offer?Ignoring commissions, the counter will buy pounds for 1.4556 dollars apiece and sellpounds for 1.4782 dollars apiece. So 1.4556 is a bid for pounds and 1.4782 is the ask oroffer for pounds. But, 1.4556 is also an offer for dollars: when the counter buys pounds,it automatically sells dollars. Similarly, the 1.4782 is also a bid for dollars.

Bank trading rooms are organized into major currency desks and cross-desks. Theformer deal in yen, euros, etc. against the dollar. The latter quote cross rates, which areFX rates directly of non-dollars against non-dollars (e.g., CHF against JPY). Emergingeconomies’ currencies are normally traded against the dollar, but cross-trading againstthe euro can be just as significant, depending on the region (e.g., PLZ/EUR). Triangulararbitrage (described in Chapter 5) ensures that cross rates stay in line with FX ratesagainst the dollar or another major. Apart from triangular arbitrage performed bycross traders and currency arbitrage desks and inventory adjustment trading due tocustomer flows through the major currency desks, the rest of trading is dominated byspeculation. Dealers interpret macroeconomic data and take positions on future FXrate changes. This is quite different from bond market dealers making profits off thebid–ask spread on customer flows. Cash bond markets are characterized by flowtrading; currency markets are distinctly speculative with less customer flow. Occasion-ally, central banks attempt to manipulate FX markets through two types of interven-tions: sterilized, where currency-for-bonds transactions offset currency-for-currencyflows, or unsterilized, with only the latter. In recent years, the central banks ofdeveloped economies have engaged in very little intervention, as the predominantmacroeconomic orthodoxy has been to allow free-floating and only to use moralsuasion (i.e., issuing deliberate public statements to be widely (mis-?) interpreted bythe markets). The central banks of non-major currencies, on the other hand, have trieda variety of other tools aimed at stabilizing their FX rates, mostly falling into the


categories of pegging or exchange controls. The most common exchange controlarrangement is to limit the amount of FX that can be exchanged at the official rateto a specified amount, sometimes tied to the level of export earnings of the owner.Often, weaker currencies are pegged to a major or a basket of major currencies with thecentral bank intervening if the market rate deviates too much from the desired targetrate. A popular arrangement in the emerging economies in the 1990s was a currencyboard (Hong Kong, Argentina, Brazil) where local currency notes were issued at a fixedratio against a pool of foreign exchange held by a monetary authority and are freelyexchangeable into it. Currency traders watch closely the performance of these and otherarrangements against the real flows of FX and take speculative positions through spotand forwards currency and money markets.

4.3 COMMODITY MARKETS

Spot commodity markets are comprised of networks of dealers who buy, store, and sellcommodities and their derivatives. Many commodities come in a variety of qualitygrades. Some are integrated with futures markets and have financial speculators inthem; some are not integrated with futures and have mostly industrial buyers andsellers. All are governed by the specifics of the production process involved in deliveringthe commodity to the market. Let us review spot trading in commodities with a viewtoward its relationship to futures markets.Metals range from precious (gold, silver, platinum) to industrial (copper, aluminum).

Some have both uses (palladium, platinum). Metals are relatively homogeneous. As aconsequence, the metals markets are liquid. They trade through a network of OTCdealers, similar to the FX market, or on some exchanges (London Metal Exchange).Petroleum product markets are dominated by producers, not dealers, and are very

complex. Products trade at many stages of the production cycle. There is an activemarket for oil and gas reserves in the ground. Participants are diversified petroleumcompanies as well as companies purely owning reserves. There is an active market forcrude oil and gas extracted from the ground. Participants are oil companies, refiners,and end-users, as well as some investment banks for speculative purposes. There arevery active markets in final products, such as gasoline, propane, and heating oil. Thepricing power for crude oil rests with the Organization of Petroleum ExportingCountries (OPEC), with Saudi Arabia playing the role of a swing producer becauseit holds 50% of world’s proven reserves and has large production overcapacity. Oilproduction and refined product delivery can be subject to frequent supply shocks.Futures markets in petroleum products can be seasonal and do reflect the conveniencevalue of owning the spot commodity.Grain markets are dominated by grain elevators who act as dealers and buy, sell, and

store grains. The futures markets serve as the benchmarks for spot prices, quoted at apremium or discount to futures, depending on the season and production conditions.Soybeans are crushed to obtain soybean oil for use as fats in human food and soybeanmeal for use as a protein supplement in animal feeds. Prices are quoted directly by theproducers of the two products.Cattle and pork trade both as livestock and livestock products. Livestock can be

traded in different stages of production. Cattle is traded at feeder weight (up to 800


pounds, roughly 1-year-old below optimal slaughter weight) or as fat cattle (�1,200pounds). Similarly, hogs trade as feeder hogs or later as market hogs. Relative prices atdifferent stages vary dependent on the cost of feeds and other factors. Both cattle andhogs are slaughtered evenly throughout the year. This renders the futures contract totrade at no convenience value.

In recent years, an active spot and futures market has developed in electricity, asgovernments throughout the world started deregulating electricity markets (Scandina-via, North America, the U.K. and continental Europe). The deregulation consisted ofseparating electricity generators from electricity transmitters and relaxing the pre-viously tight price controls. Often, governments have been unwilling to give up someprice controls. The result has been a lively and sometimes chaotic market for spotelectricity. The product itself is uniform (there are no quality grades), but it is notstorable. This automatically means that the generators must maintain an overcapacityto be able to deliver during peak times. This also means that demand and supplyconditions during peak and off-peak times are completely different. Buyers andsellers must engage in what is called load-matching. Any delivery contract is extremelyspecific in terms of the quantities to be delivered at different times of the day, differenttimes of the week, and the number of days per month guaranteed for on- and off-peak.In many cases, the arrangements are complicated by obsolete and overly regulateddelivery networks. Most electrical grids suffer from natural bottlenecks (California–Oregon Border, or COB and Palo Verde, Arizona, or PV) and man-made bottlenecks(New Jersey–Pennsylvania separated from the Midwest by regulation and nationalgrids in Europe). All of these factors contribute to the very high volatility of electricityprices (and recently a few, large blackouts in the U.S. and Europe). While the long-termcost of generation depends greatly on the price of fossil fuels (for the most part oil andgas), the short-term supply conditions often lead to great deviations from that.

In later chapters, we show how the spot prices of commodities are linked to futuresprices through the cost of carry and convenience yields. We also cover more interestingaspects of electricity futures trading.


___________________________________________________________________________________________________________________________________________________________________________ 5 __________________________________________________________________________________________________________________________________________________________________________

_____________________________________________________________________ Spot Relative Value Trades ____________________________________________________________________

Relative value trades in spot markets are relatively simple in concept. They rely onspeed of execution as they often take advantage of temporary mispricing of assetsrelative to each other. They are self-correcting in that the very transactions thatexploit mispricings also eliminate them by pushing prices in line with each other. Inmany instances, their success depends on the existence of lesser informed traders orinstitutional and retail customers who are less price-sensitive. Some are pure arbitrages,but more often than not they rely on relative value principle and result in smallsecondary risks, like yield curve (YC) exposure, commodity basis risk, or market-neutral sector risk. We review them not in the order of simplicity, but rather in thesequence of the last two chapters.

5.1 FIXED INCOME STRATEGIES

We start with relative value trades which exploit the fact that coupon securities can beviewed as packages of zero-coupon securities and the whole must be equal to the sum ofthe parts (in terms of the present value, or PV, of the components).

Zero-coupon stripping and coupon replication

Zero-coupon stripping and coupon replication are most widely used in governmentbond markets around the world. Corporate bond traders with access to securitylending may also engage in coupon replication, but limit the application of this strategyto the highest rated bonds of very large issuers because they may be additionallyexposed to default or corporate spread risk.The most basic zero-vs.-coupon arbitrage is executed in the U.S. Treasuries and

STRIPs (separated coupons or principals off original Treasuries sold as individualsecurities) markets. The issuer of all securities is the same, the U.S. government. Allsecurities are default-free and easily marginable (i.e., both coupons and zero-couponscan be shorted). Let us examine how we would screen for the possibility of arbitrageand how we would execute riskless trades to earn sure profit. We assume none of thesecurities is on special (i.e., no rebate is charged for borrowing and shorting through arepo; in other words, repoing). Consider the following set of quotes (semi-annual bondequivalents on a 30/360 basis, prices in decimals) for STRIP (zero) yields and couponbond prices:

Table 5.1 Zero-coupon and coupon bond rates and prices

Maturity Zero Coupon rate Coupon price(years) (yield)

0.5 3.5000 3.50 100.00001 3.5931 3.60 100.00731.5 3.7281 3.75 100.03622 3.7903 3.80 100.02632.5 3.8817 3.85 99.76423 4.1386 4.00 99.6646

From the coupon bond prices and given the coupon rates, we can compute the yields-to-maturity (YTMs) implied for the six coupon bonds following the formulae ofChapter 2. YTMs are defined as the single yields used to discount the cash flows ofeach bond to get the price (zero yields are not used in these calculations). We add theyield column to the table of quotes:

Table 5.2 Zero-coupon and coupon bond rates, prices, and yields

Maturity Zero Coupon(years) (yield) ——————————————————————

Rate Price Yield

0.5 3.5000 3.50 100.0000 3.50001 3.5931 3.60 100.0073 3.59251.5 3.7281 3.75 100.0362 3.72502 3.7903 3.80 100.0263 3.78632.5 3.8817 3.85 99.7642 3.95003 4.1386 4.00 99.6646 4.1200

We can also discount the cash flows of each coupon bond by applying the appropriatezero yields to the coupon cash flows and principal cash flows to compute the theoreticalprice and then the theoretical YTM of each coupon bond, assuming the zeros are pricedcorrectly (zero yields used). We replace the ‘‘Price’’ and ‘‘Yield’’ columns in the tablewith their theoretical equivalents:

Table 5.3 Theoretical prices and yields of coupon bonds computed by discounting cash flowswith zero-coupon rates

Maturity Zero Coupon(years) (yield) ——————————————————————

Rate Price Yield

0.5 3.5000 3.50 100.0000 3.50001 3.5931 3.60 100.0073 3.59251.5 3.7281 3.75 100.0362 3.72502 3.7903 3.80 100.0263 3.78632.5 3.8817 3.85 99.9410 3.87503 4.1386 4.00 99.6646 4.1200

We notice that the theoretical price and yield of the 2.5-year 3.85% coupon note is


different from that actually quoted in the market. The 2.5-year trades cheap. Its price islower than what it should be relative to the zeros; or, it may be that one of the zeros,specifically the 2.5-year one, trades rich relative to the coupon. We are not making astatement as to which is overpriced and which is underpriced by some absolute stan-dard. The theoretical prices we computed were strictly for the purpose of discoveringarbitrage.One correct strategy in this case will involve buying the 2.5-year coupon and shorting

five different STRIPs (zeros) ranging in maturity from 0.5 to 2.5 years. The face value ofeach zero shorted would be equal to 1.925 (semi-annual coupon of 3.85/2), except forthe last one whose face value would be 101.925 per 100 face value of the couponbought. The amounts are summarized in Table 5.4:

Table 5.4 Proceeds from shorting five STRIPs with face values matching the cash flows of the2.5-year coupon bond

Maturity Zero Face value Price received(years) (yield)

0.5 3.5000 1.925 1.89191 3.5931 1.925 1.87771.5 3.7281 1.925 1.82122 3.7903 1.925 1.78572.5 3.8817 101.925 92.5847

Total 99.9412

The total amount received from the shorts is greater than the price paid for the 2.5-yearmaturity 3.85% coupon, by 99:9412� 99:7642 ¼ 0:1770, or 17.7 basis points. On a$100 million face value trade, this amounts to $177,000. There is no risk to the tradeas the received coupon payments will cover exactly the return of the zero principals tothe lenders of the securities. The only risk may be call risk if one of the STRIP shortscannot be rolled over in an overnight repo market.This is not the only arbitrage strategy that can be executed based on the quotes:

another may be to buy the 2.5-year, 3.85 coupon, sell a shorter coupon, and buy andsell some zeros. For example, we can short the 2-year 3.80 coupon, combined with smallshorted strips of 0.5- to 2-year zeros (with principal equal to $3.85 million minus $3.80million divided by 2, i.e., $25,000), short a $101.925 million 2.5-year STRIP, and buy a2-year $100 million STRIP to offset the principal of the 2-year coupon note. This willwork because all zeros and coupons with maturities equal to or less than 2 years in ourexample are fairly priced relative to each other. So, they are costless substitutes. Theonly consideration for the trader is the availability of the components and the total bid–ask spread paid to execute all the transactions.The coupon-stripping strategy is an example of a static strategy. Once all the initial

transactions are executed, the trader simply collects the coupons and principals fromthe longs when they come due and allocates them to the shorts. All future cash flowsmatch up exactly. There are no additional trades to be done. The profit is made upfrontfrom the difference between monies received from the shorts and monies paid on thelongs. We can simply wait for the last cash flow to consider the profit fully realized. Theonly time we may consider unwinding the entire trade would be if, during the life ofthe strategy, the total mark-to-market on the strategy, net of additional transaction

Spot Relative Value Trades 115

costs, becomes positive, due to new relative mispricing of the components. That can bea source of additional profit.

Duration-matched trades

Another way of arbitraging spot bond markets is through dynamic strategies of dura-tion matching. In some cases, the secondary speculative component is the difference inbond convexities if one intends to profit from parallel yield curve movements. Moreoften, convexities may also be matched and profit is earned as duration- and convexity-matched bond positions adjust differently to non-parallel yield curve movements.(Chapter 2 offers definitions and explains the economic meaning of duration andconvexity.) Typically, the relative arbitrageur selects a primary bond (or bonds)which is bought/shorted and hedges it by shorting/buying other bonds with the dura-tion of the hedges equal in absolute value to that of the primary bond. This ensures thatthe speculation is not directly on the level of interest rates. The trader profits or losesdue to parallel movements through a convexity mismatch (second-order variable) and/or due to non-parallel yield curve movements even if the convexities are matched. Thestrategy is dynamic. The cash flows on the trade positions do not match. The traderdoes not intend to hold the positions to maturity, but only until he realizes a desiredlevel of profit. He dynamically adjusts the duration match as interest changes.

Example: Bullet–barbell

Let us illustrate with a bullet–barbell combination, originally designed as a hedgestrategy aimed at immunizing against parallel yield curve movements. Here we showhow a duration- and convexity-matched bullet–barbell strategy profits from non-parallel movements.

Suppose we buy $116.460 million of a 5-year note with the modified duration of 3.22and convexity of 0.33. That is, a 1 bp (basis point) increase in yield will cause a 3.22 bpdecrease in value using the straight-line approximation minus a 0.33 bp correction for acurvilinear quadratic approximation term. At the same time, we short a barbell (twobonds, short-maturity and long-maturity): a $48.885 million 2-year note with the dura-tion of 1.68 and convexity of 0.12, and a $42.507 million 10-year note with the durationof 6.89 and convexity of 0.76. As can be seen from Table 5.5, the face value amountshave been chosen so that the portfolio is duration-neutral and close to convexity-neutral.

Table 5.5 Long 5-year bullet, short 2–10 barbell

Bond Face value Duration Convex $ Duration $ Convexityper bp per bp

2-year (48,885,000) 1.68 0.12 8,213 (587)5-year 116,460,000 3.22 0.33 (37,500) 3,84310-year (42,507,000) 6.89 0.76 29,287 (3,231)

Total (0) 26

The 2-year bond position will gain (lose) $8,213 per 1 bp increase (decrease) in the


YTM, the 5-year will lose (gain) $37,500 per 1 bp increase (decrease) in the YTM, andthe 10-year will gain (lose) $29,287 per 1 bp increase (decrease) in the YTM. Overall, ifall three YTMs move (up or down) in parallel by 1 bp, or a few bp, the portfolio will notmake or lose money. (Over a larger movement in yields, the portfolio will make a tinyamount of money as it is lightly convex.)The portfolio will profit/lose from non-parallel yield curve movements. If the yield

curve steepens (i.e., the 2–10-year yield curve differential increases by 10 bp), theportfolio will gain over $105,000. If the yield curve flattens by 10 bp, the portfoliowill lose roughly the same amount. The portfolio will also profit or lose, if the yieldcurve twists. For example, if the 2-year and 10-year yields decline by 5 bp, but the 5-yearyields decline by 10 bp, the portfolio will gain $187,501. The two scenarios are summar-ized in Table 5.6 which shows the sources of profits in each case.

Table 5.6 Long 5-year bullet, short 2–10 barbell—profit from yield curve (YC)

Bond Duration YC steepens YC twists————————————— —————————————Move (bp) $ Profit Move (bp) $ Profit

2-year 1.68 5 41,063 �5 (41,063)5-year 3.75 10 (375,001) �10 375,00110-year 6.89 15 439,310 �5 (146,437)

Total 105,372 187,501

Example: Twos vs. tens

One of the simplest and most common strategies to benefit from non-parallel yieldcurve movements is a duration-matched twos–tens trade in which the arbitrageurgoes long 2-year notes and short 10-year notes, if he intends to profit from a steepeningof the yield curve, or vice versa (i.e., short 2 years and long 10 years) if he intends toprofit from a flattening. In this strategy, it is much more difficult to eliminate convexionrisk, so there is normally some residual primary interest rate risk.Suppose we want to profit from a yield curve flattening. We short $61.3 million of 2-

year notes and we buy $14.947 of 10-year notes. We choose the amounts to eliminateinterest rate risk (i.e., we match the duration of the short and the long).

Table 5.7 Short twos–long tens


2-year (61,300,000) 1.68 0.12 10,298 (736)10-year 14,947,000 6.89 0.76 (10,298) 1,136

Total (0) 400

If all yields (2-year and 10-year) rise in parallel by 10 bp, then the 2-year position willgain close to $103,000 and the 10-year position will lose the same amount. However, ifthe yield curve flattens (i.e., 2-year yields rise by more, or drop by less, than 10-year


yields), then the strategy will make a profit. In Table 5.8, 2-year yields rise by 10 bp,while 10-year yields rise only by 5 bp, resulting in a 5 bp flattening. The strategy gains$51,492. At this point, the trader may choose to close out both positions. If an evengreater flattening is anticipated, the trader may choose to rematch the durations orleave them slightly mismatched to ride the strategy longer.

Table 5.8 Short twos–long tens

Bond Duration YC moves parallel YC flattens————————————— —————————————Move (bp) $ Profit Move (bp) $ Profit

2-year 1.68 10 10,984 10 102,98410-year 6.89 10 (102,985) 5 (3,51,492)

Total (1) 51,492

If both yields have risen, then the durations of both bonds have decreased. Given theoriginal convexity numbers, the duration of the 2-year notes will have changed to 1.662and that of the 10-year to 6.852. Duration rematching can be accomplished, forexample, by shorting additional $0.101 million 2-year notes and leaving the 10-yearposition unchanged.

Table 5.9 Short twos–long tens—duration rematch

Bond Face value Duration $ Durationper bp

2-year (61,401,000) 1.668 10,24210-year 14,947,000 6.852 (10,242)

Total 0

Another yield curve flattening of 5 bp will now result in a profit of $51,208, while aparallel move will result in no value change.

Negative convexity in mortgages

Because of the prepayment risk, the management of fixed-rate mortgages and mort-gage-backed securities (MBSs) is particularly risky. A duration-matched portfoliomakes or loses money not only for large movements in yields, but sometimes forsmall parallel movements when yield curve movement leads to an acceleration ordeceleration or prepayments. The change in the speed of prepayments causes thedurations to become mismatched and the interest rate sensitivities of the variousbonds in the portfolio to diverge. Mortgage bond durations are calculated for agiven speed of prepayments. The latter is measured in PSAs. (PSA stands for thePublic Securities Association, but it is commonly used as a unit of prepaymentspeed, as defined by the Association.). As current PSAs change, so do the durationsand interest rate sensitivities. In addition, the changes are not symmetric. Downwardmovements in the yield curve may accelerate prepayments by more than upward move-


ments will slow them down. There are many reasons for such a situation. One com-monly considered factor is the past history of curve movements. If interest rates havenot changed much in the past and started coming down only recently, a large wave ofrefinancings (and thus prepayments) may be unleashed. On the other hand, if interestrates have been coming down rather steadily in the past, a further drop may actuallylead to a slowdown in prepayments as there may now be fewer mortgages to berefinanced. These issues are most acute with highly structured MBSs (like IOs, orinterest-onlys, and POs, or principal-onlys) and are the main focus of statistical pre-payment modeling done by most mortgage-dealing firms.The crux of the matter is the potential negative convexity of fixed-rate mortgages. For

a non-mortgage bullet bond, as interest rates come down the price and the duration ofthe bond increase. For a mortgage bond, the price may increase but the duration maydecrease. To understand that recall from Chapter 2 that one interpretation of durationis the present value-weighted average time to the bond’s cash flows. For bullet bonds, asrates decrease, the present value of far cash flows increases relative to near ones. ThePV-weighted average time to the repayment of the bond (i.e., the duration) increases. Inthe extreme, when rates are zero, the duration is at a maximum and it is a cash flow-and not PV-weighted average time measure. Conversely, as interest rates rise, the PV offar cash flows including the principal, if any, declines more relative to the near cashflows and duration contracts. In the extreme, when rates go infinite, duration is zero.The same intuitions apply to mortgages to the extent that they are coupon-paying,

fixed-income instruments and prepayment speeds do not change. As interest ratesdecline, the PV of a given far cash flow from a mortgage increases more than that ofa near one. But, if the prepayment speed increases, there may be fewer far cash flowsand more near cash flows. The overall duration of the bond, instead of increasing, mayactually decline. Consider a bond backed by a ‘‘pool’’ of two identical 15-year mort-gages. As interest rates decline, the PV of the cash flows scheduled for years 10–15increases more than that of the cash flows scheduled for years 1–5 (due to the power ofcompound interest). The PV-weighted average time to the repayment of the mortgagebond increases. But now suppose that the owner of the second mortgage decides to paydown her principal in order to avoid paying a now high interest rate. This moves the farcash flows to today. The PV-weighted average time for the entire pool declines; so doesthe sensitivity of the price of the pool-backed bond to interest-rate changes. Theopposite is true for an interest rate increase. In that case, prepayments slow down,extending the bond’s duration even though the PVs of far cash flows decline. Theextreme case of this negative convexity is found in PO STRIPs of MBSs where theowner is entitled to the principal portion of monthly mortgage payments coming fromthe pool. As rates decline, a PO holder may be swamped with principal prepayments,practically extinguishing his bond and exposing him to the reinvestment risk at the nowlower interest rates.Some structured MBSs exhibit negative duration. The owner of the IO STRIP is

entitled to the interest portion of mortgage payments. As rates decline and prepaymentspeeds increase, promised future interest never materializes and the PV of the cash flowsdecreases.Mortgage bond portfolio managers are normally long on mortgage bonds and short

on hedges. The hedges include government or agency bonds. The two sides can beduration-matched in an attempt to immunize against explicit interest rate risk. But


duration matching is quite tricky in this case. At a given assumed prepayment speed,the portfolio may appear duration-neutral. But long mortgage bonds (negativelyconvex) combined with short governments (positively convex when owned, but hereshorted, hence negatively convex) may lead to particularly short-convexity portfolios.These make money if PSAs do not change, but lose a lot of money when they do.Mortgage traders may resort to callable bonds or swaptions as hedges. Buying call swapoptions offsets the options implicitly sold to the homeowners in the pool. The hedgemay be the best we can do, but may be highly imperfect as it depends on future ratelevels, and not their histories or other macroeconomic and personal factors leading tomortgage refinancings.

The negative convexity of mortgages can be used to our advantage in a variety ofyield curve strategies. Consider a twos–tens curve-flattening strategy. Suppose in addi-tion to the 2-year and 10-year government notes we include in the portfolio somemortgage bonds backed by a pool of 30-year mortgages that are currently prepayingat some PSA rate such that the duration of these bonds is close to that of the 10-yeargovernment note. Their convexities, instead of being very similar to those of the 10-yearnotes, are actually negative because of the negative prepayment effect. Instead of short-ing 2-years and buying 10-years, we short 2-years, buy 10-years, and buy mortgagebonds with face amounts chosen to zero out the overall convexity, as in Table 5.10.

Table 5.10 Short twos–long tens and mortgages


2-year (61,300,000) 1.68 0.12 10,298 (736)10-year 10,600,000 6.89 0.76 (7,303) 806Mortgages 4,524,000 6.62 �0.15 (2,995) (68)

Total — 2

We are still immunized to parallel yield curve movements, and we still make the sameamount of $51,493 on a 5-bp yield curve flattening, but we have no exposure to non-parallel movements.

Table 5.11 Short twos–long tens and mortgages: Hedge performance

Bond Duration YC moves parallel YC flattens———————————— ————————————Move (bp) $ Profit Move (bp) $ Profit

2-year 1.68 10 102,984 10 102,98410-year 6.89 10 (73,034) 5 (36,517)Mortgages 6.62 10 (19,949) 5 (14,974)

Total 1 51,493

We do have rebalancing to do if we want to continue the flattening strategy with a


duration match. Convexity will continue to be close to 0. The summary of the rebalan-cing statistics is shown in Table 5.12.

Table 5.12 Short twos–long tens and mortgages—duration rematch

Bond Duration $ Duration $ Convexityper bp per bp

2-year (61,520,000) 1.668 10,262 (738)10-year 10,600,000 6.852 (7,263) 806Mortgages 4,524,000 6.6275 (2,998) (68)

Total 0 (0)

Spread strategies in corporate bonds

Corporate bond trading at broker-dealer firms has two components: inventory tradingdue to customer flows and spread arbitrage. The first is similar to government bondtrading and focuses on the bid–ask spread for the more liquid issues. As customers callwith buy and sell orders, dealers trade out of their inventory, earning profit simply bycharging more than paying for the same bonds, or their close substitutes (e.g., sameissuer, different maturity, and coupon). The line between flow trading and spreadarbitrage gets blurred as the definition of a substitute is expanded to include bondsof other issuers, but of the same credit quality. Dealers then accumulate bonds of onetype and are short of another, staying neutral with respect to interest rate risk. Thiseither happens naturally as flows become unbalanced or it is by design in order tobenefit from spread movements. Let us give some examples of the latter.

Example: Corporate spread widening/narrowing trade

Suppose we are a corporate bond dealer and we believe that the spread between A-ratedbonds and government securities is going to narrow. We believe that this will be aconsequence of the sparsity of new corporate issuance which will limit the supply ofthe bonds. Our view on interest rates overall is neutral.In order to benefit from the anticipated narrowing of the spread, we can buy A-rated

corporate bonds and short governments with the same maturity. We match the dura-tions of the long and the short by choosing the face amounts appropriately. This issummarized in Table 5.13.

Table 5.13 Long corporate, short government


10-year government (45,210,000) 6.89 0.76 31,150 (3,436)10-year A-rated corporate 50,000,000 6.23 0.56 (31,150) 2,800

Total (0) (636)


Let us consider what will happen to the position as the spread between the yield on theA-rated bond and the government bond narrows. This can occur in a variety of ways.One scenario is that both yields rise, but the A yield rises by less than that of thegovernment bond. Suppose the respective yield changes are 30 bp and 50 bp, resultingin a narrowing of 20 bp.

Table 5.14 Long corporate, short government—rates rise by 30 bp (corporate) and 50 bp(government)

Bond Duration Yield move (50 bp) Spread move (�20 bp)——————————— ——————————Move (bp) $ Profit Move (bp) $ Profit

10-year government 6.89 50 1,557,485 0 —10-year A-rated corporate 6.23 50 (1,557,500) �20 623,000

Total (16) 623,000

The change in the value of the portfolio due to the overall interest rate increase is nil.This result is ensured by the duration match. There is, however, a substantial profit of$623,000 which is due to the narrowing of the spread.

This strategy can be combined with a small outright bet on rates to form a speculativeposition. Spread widening often occurs during broad rate increases, and narrowingduring rate declines. Suppose we want to bet on the scenario that the spread narrowingis going to follow a general rate decline. We can mismatch the durations slightly toleave ourselves net long. We can short fewer government bonds.

Table 5.15 Long corporate, short government—mismatch


10-year government (40,000,000) 6.89 0.76 27,560 (3,040)10-year A-rated corporate 50,000,000 6.23 0.56 (31,150) 2,800

Total (3,590) (240)

This will earn additional profit if rates overall decline, but will hurt our profit if ratesincrease. We still benefit from the narrowing of the spread.

Table 5.16 Long corporate, short government—spread narrows

Bond Duration Yield move (50 bp, 30 bp) Yield move (�30 bp, �50 bp)——————————— ——————————————Move (bp) $ Profit Move (bp) $ Profit

10-year government 6.89 50 1,378,000 �30 (826,800)10-year A-rated corporate 6.23 30 (934,500) �50 1,557,500

Total 443,500 730,700


If the spread narrows by 20 bp while yields rise, our profit will shrink from $623,000 to$443,500. If the spread narrows while yields decline, our profit will increase to $730,700.The difference between the new profit levels and the original matched strategy is all dueto net duration exposure to interest rates.

Example: Corporate yield curve trades

The duration-matched spread widening/narrowing strategy can be considered specula-tive. The bet is not outright on interest rates, but on the corporate spread which can behighly volatile at times, resulting in large profits or losses on the government-hedgedcorporate portfolio.A finer trade is a bet on the spread between different maturity points of the corporate

spread for a given category of issuers. In this case, the dealer is interest rate-neutral andcorporate spread-neutral, but speculates that the difference between the corporatespread for 10-year maturities and 2-year maturities will change.Suppose that currently the spread between A-rated issues and governments is 30 bp

for 2-year maturities and 70 bp for 10-year maturities. We believe the 40-bp difference isgoing to increase, reflecting improved default prospects in the short run relative to thelonger run. We do not have a view on interest rates in general or corporate spreads ingeneral. This time we will match durations of corporates and governments for eachmaturity segment and across the segments. This is accomplished in the following way.

Table 5.17 Corporate spread steepening trade


2-year government (166,875,000) 1.68 0.12 28,035 (2,003)2-year A-rated corporate 183,235,000 1.53 0.13 (28,035) 2,38210-year government 40,700,000 6.89 0.76 (28,042) 3,09310-year A-rated corporate (45,000,000) 6.23 0.56 28,035 (2,520)

Total (7) 953

The strategy is immune to parallel and non-parallel yield curve movements. It is alsoimmune to parallel corporate spread movements. It only makes money if the differencebetween the 10-year and 2-year spread increases (i.e., the corporate spread curvesteepens). It loses the same amount if the difference decreases.

Table 5.18 Corporate spread steepening trade

Bond Duration 0 bp steepening 20 bp steepening——————————— —————————————Move (bp) $ Profit Move (bp) $ Profit

2-year government 1.68 10 280,350 10 280,3502-year A-rated corporate 1.53 10 (280,350) 10 (280,350)10-year government 6.89 20 (560,846) 20 (560,846)10-year A-rated corporate 6.23 20 560,700 40 1,121,400

Total (146) 560,554


In Table 5.18, a yield curve steepening of 10 bp results in no profit, but the same yieldcurve steepening combined with a steepening of the corporate spread curve of 20 bpresults in a profit of $560,554.

Example: Relative spread trade for high and low grades

The spread between low investment grade and high investment grade issues can be quitevariable. In order to benefit from the relative value trades between these two categorieswe can design trades similar to the strategies described in the previous two subsections,where the role of the government bond is played by the high-grade category. Thephilosophy of the trade is the same. We show one example of such a trade.

Suppose we believe that the spreads between low-grade and high-grade issues willnarrow and that the spread for the 10-year maturity point is going to narrow more thanthat for the 2-year point. We have no view on government yields, the spreads betweenhigh grade issues and governments, or the 10-year–2-year spread for each categoryalone.

Table 5.19 Low-grade–high-grade spread trade


2-year AAA-rated corporate 149,196,000 1.68 0.12 (25,065) 1,7902-year A-rated corporate (183,235,000) 1.53 0.13 28,035 (2,382)10-year AAA-rated corporate (45,000,000) 6.89 0.76 31,005 (3,420)10-year A-rated corporate 45,000,000 6.23 0.56 (28,035) 2,520)

Total 5,940 (1,492)

We mismatch durations by about the same amount for each maturity category. The sizeof the mismatch reflects our view on overall tens–twos spread narrowing, and we golong 10-year relative spreads and short 2-year relative spreads. We make money whenthe relative spread between As and AAAs narrows and the 10-year spread narrowsmore.

Table 5.20 Low-grade–high-grade spread trade

Bond Duration 30 bp A–AAA-rated þ10 bp flatteningnarrowing

—————————— ———————————Move (bp) $ Profit Move (bp) $ Profit

2-year AAA-rated corporate 1.68 30 (751,948) 30 (751,948)2-year A-rated corporate 1.53 0 — 0 —10-year AAA-rated corporate 6.89 30 930,150 30 930,15010-year A-rated corporate 6.23 0 — �10 280,350

Total 178,202 458,552

When relative spreads narrow by 30 bp across the maturity spectrum, we make


$178,202. When, in addition, the 10-year spread narrows by 10 bp more (i.e., 40 bp), wemake $458,552.

5.2 EQUITY PORTFOLIO STRATEGIES

Benchmarking in equity markets commonly refers to strategies aimed at outperforminga stock index. The performance of a portfolio is measured in terms of excess returnsover those of the index. The objective is not necessarily to have positive returns, but tohave the largest excess return (the index itself could have a negative return) over a targethorizon (i.e., the objective is to beat the index).Market-neutral hedge funds often go long a selected portfolio and short the index

portfolio (in cash or futures markets) in order to make an explicit outperformanceprofit. If the underlying long portfolio is closely related to the index, then sometimesit is cheaper to net all individual longs against the shorts and to long/short the netpositions. The market-neutral hedge fund can be highly leveraged as the value of longsand short is approximately equal and the capital required to maintain the position isminimal.Lacking a concept like duration in fixed income markets, in equities relative value

arbitrageurs rely on statistical measures of market exposure to match longs against realor benchmarked shorts. This is accomplished with the use of a ‘‘market model’’ whereeach stock is considered as having exposure to a set of factors. In the classic capital-asset pricing model (CAPM),1 familiar from college finance textbooks, there is only onesuch factor and it is the market index itself. The benchmark market portfolio has afactor loading of 1, and the stocks’ loadings (betas) are estimated using a regression onmarket returns.Suppose we compute the monthly returns on MSFT and on the S&P 500 index for

the period of Jan-1995 through Dec-2002 and plot them on a scatter diagram, as inFigure 5.1. Each dot represents the returns for MSFT and S&P 500 for a given month.Also suppose we then run a linear regression of MSFT on S&P 500 to fit a straight linethrough the scatter. The slope of the line is the beta of MSFT and is supposed to tell ushow MSFT stock will perform as a function of S&P. For the 95-02 period, the monthlyreturn on MSFT is related to the monthly return on S&P 500 through the followingequation of the line:

~rrMSFT ¼ 0:915 599%þ 1:588 493 � ~rrS&P500

As can be seen from the plot, the historical data points do not lie on a straight line;rather, they are scattered around the line defined by the regression results. This meansthat MSFT returns may be driven by factors other than the general uncertainty ofstocks portrayed by the market index.A California company, BARRA, has popularized a more complicated, macro-

economic factor-based model, in which each stock’s and the index’s return are functionsof several factors (stock market index, spread between 2-year and 10-year bond, realestate index, etc.), allowing us to come up with factor loadings through a multipleregression (a subplane in a multi-dimensional scatter space). In a more abstract


1 See any college finance text for a CAPM exposition. For example, Richard A. Brealey, Stewart C. Myers, and Alan J.Marcus, Fundamental of Corporate Finance (4th edn), 204, McGraw-Hill Irwin, Chicago.

Arbitrage Pricing Theory, or APT, model, factors are defined only implicitly, but theloadings can be estimated through a principal component decomposition. Most marketmodels relate the return on any stock ~rr to a set of k factors and can be written as:

~rr ¼ b0 þ b1 ~ff1 þ b2 ~ff2 þ � � � þ bk ~ffk þ ~""

Each stock has a unique set of factor loadings, ðb0, b1, b2, . . . , bkÞ, which can beestimated from historical data. Once we know the set of returns on the factors,ð ~ff1, ~ff2, . . . , ~ffkÞ, we automatically know the return on each stock. Over time, theerrors from this assumption are assumed to cancel out, E½~""� ¼ 0.

The role of the model in all cases is to estimate the co-movement of the portfolio withthe benchmark, similar to the way duration predicts the price change of the bond as aresult of a yield change. Here one statistic is not enough. For each stock, unique factorloadings are used to predict the change in the price of the stock as a function of thefactor changes. Similarly, the factor loadings for the benchmark are used to estimatethe change in the benchmark as a function of the factors. This way of approaching theproblem ensures that we have some ex ante measure of how the portfolio is related(matched or mismatched) to the benchmark. It also allows quantification of the com-parison of two completely different portfolios to each other and to a benchmark interms of factor overloading and underloading.

Example: A non-diversified portfolio and benchmarking

Suppose we are managing a portfolio consisting of 15 U.S. stocks. (There are manymutual funds specializing in 15- to 30-stock portfolios.) The portfolio is somewhatdiversified to eliminate large specific risks (exposures to single stocks), but much lessthan a broad market index. Suppose our performance is judged against the S&P 500index.

Let us use a model, stylized on the Fama–French2 setup popular in academic


-20%

-15%

-10%

-5%

0%

5%

10%

15%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40%

S&P

MSFT

Figure 5.1 MSFT vs. S&P 500 monthly returns.Source of data: Yahoo!Finance.

2 Eugene F. Fama and Kenneth R. French, ‘‘Common risk factors in the returns on stocks and bonds’’, Journal of FinancialEconomics, February, 1993, 33(1), 3–56.

literature, to explain long-term stock returns. Three factors are assumed to affect thevariability of a stock’s return: the market index, firm size (small is better) and book-to-market value (high book/market value ratio is better), both relative to the average. Thefirm size factor is defined as the return on a zero-net-investment portfolio long on smalland short on large stocks. The book-to-market factor is defined as the return on a zero-net-investment portfolio long on high book-to-market (value) and short on low book-to-market (growth) stocks. The S&P 500 index has factor loadings of (1, 0, 0). The S&Preturns vary one for one with the first factor. They are unrelated to the second andthird, as the last two are defined in terms of the average and the S&P is the average. Astock with factor loadings (0.88, 0.12, �0.08) varies less than one-for-one with the S&P500, is positively correlated with small firms’ returns and negatively correlated with highbook-to-market value stocks. In any given period, knowing the realizations of thefactors, we are able to compute the forecast returns of the 15 stocks in the portfolioand the benchmark index. The model is used to construct the equivalent of a durationmatch, by picking stock in such proportions as to eliminate the exposure to all threefactors in a market-neutral strategy or to match the factor loadings of the benchmarkportfolio in a benchmark outperformance strategy.Table 5.21 contains an example of a $16,200,000 portfolio of 15 stocks benchmarked

against the S&P 500 index. With 15 stocks, there are many ways to eliminate exposureto the three factors. We choose the proportions in each stock, so that the net dollarexposure to each factor matches that of the benchmark: in this case (1, 0, 0).

Table 5.21 Long a 15-stock portfolio—Short S&P 500 index. Stock amounts chosen to eliminatefactor exposures

Stock Factor loadings $ Position $ Exposure Profit from factor move————————— ——————————————— ————————————1 2 3 1 2 3 5.00% 1.00% �2.00%

1 0.78 0.12 �0.22 1,800,000 1,404,000 216,000 (396,000) 70,200 2,160 7,9202 0.81 0.07 �0.08 1,700,000 1,377,000 119,000 (136,000) 68,850 1,190 2,7203 1.17 0.02 0.06 1,600,000 1,872,000 32,000 96,000 93,600 320 (1,920)4 1.21 �0.04 0.20 1,330,000 1,609,300 (52,500) 266,00 80,465 (525) (5,320)5 1.25 �0.08 0.18 1,400,000 1,750,000 (112,000) 252,000 87,500 (1,120) (5,040)6 0.84 0.11 0.07 1,300,000 1,092,000 143,000 91,000 54,600 1,430 (1,820)7 0.87 0.04 �0.08 1,200,000 1,044,000 48,000 (96,000) 52,200 480 1,9208 0.90 �0.07 �0.15 1,150,000 1,035,000 (80,500) (172,500) 51,750 (805) 3,4509 1.07 �0.10 0.11 500,000 535,000 (50,000) 55,500 26,750 (500) (1,110)10 1.21 �0.17 0.18 900,000 1,089,000 (153,000) 162,000 54,450 (1,530) (3,240)11 1.34 0.33 �0.10 800,000 1,075,700 264,00 (80,000) 53,785 2,640 1,60012 1.49 0.11 �0.15 700,000 1,043,000 77,000 (105,000) 52,150 770 2,10013 0.93 �0.11 0.14 600,000 558,000 (66,000) 84,000 27,900 (660) (1,680)14 0.64 �0.33 0.11 500,000 320,000 (165,000) 55,000 16,000 (1,650) (1,100)15 0.99 �0.55 �0.19 400,000 396,000 (220,000) (76,000) 19,800 (2,200) 1,520

Index 1.00 0.00 0.00 (16,200,000) (16,200,000) — — (810,000) — —

Total — (0) — — (0) 0

We show that for any move in the market the portfolio is immunized (i.e., it willperform just like the index). Suppose the market overall moves up by 5% over theinvestment horizon. At the same time, small stocks outperform large by 1% and thegrowth stocks outperform value stocks by 2%. Our portfolio gains $810,000 which


matches the 5% index performance and we have no exposure to the firm size or book-to-market value factors.

Example: Sector plays

There are a few commercial providers of return databases and optimization softwareaimed at performing portfolio selection in a factor-based model (e.g., BARRA,CRISP). These can be used in benchmark matching or other quantitative arbitragestrategies. These techniques may also be combined with and aid in the fundamentalstock research within an asset management company. In that case, we use the statis-tically obtained factor loadings to choose investments to immunize against marketfactors, but opt to reallocate the thus-prescribed amounts in favor of positively eval-uated stocks (through fundamental analysis). The resulting mismatch in the loadingsprovides a measure of exposure to the risk factors. The optimization software allows usto select portfolios not only with target exposures to the underlying predefined riskfactors, but also with a particular relationship to other portfolios (e.g., industrialsectors). This is accomplished through what is called factor rotation. The logic is thefollowing. If we know the exposure of our portfolio to the factors and we know theexposure of another portfolio to the factors, then we are able to determine the necessaryreallocation of our portfolio to obtain the same exposure to the factors as that of theother portfolio. If the comparison portfolio is identical to an industrial sector or abenchmark, we are able to define our portfolio in terms of the exposures to thatsector or benchmark, and not only the factors.

Suppose we want to construct a 15-stock portfolio whose exposure is roughly equal-weighted to sector 1 (telecom), sector 2 (semiconductor), and sector 3 (software). Theeasiest way to do this is to use a factor model, define each sector in terms of factorexposures, and match the factor exposure of our portfolio to that of the sectors. Asthere are 15 stocks and 3 factors, there are many solutions to this problem. Often weinsist that certain stocks be included and allocated some dollar minimum (e.g., forliquidity reasons). One solution is given in Table 5.22. In principle, this is identical tothe benchmarking exercise. The procedure allows us to claim that our portfolio has thesame performance profile as a portfolio with weights 50/162 in sector 1, 58/162 in sector2, and 54/162 in sector 3. Instead of in terms of macroeconomic factors, we are able todefine risk in more familiar terms: telecom, semiconductor, and software. Notice thatwe can define the target any way we want. For example, the target can be long telecom–short semiconductor; or we can define the target as long S&P 500 with 10% overweightin telecom and 10% underweight in semiconductor. The method is quite general in thatwe can construct sector spread portfolios, underweighted portfolios, etc. All we need todo first is to decompose the target into the factor exposures and then match thosethrough optimized stock selections.

One word of caution is that equity models rely on statistics. Models based ondifferent factors may come up with different proportions of the same stocks, giventhe same target. This is the main difference between the statistics of equity approachesand the mathematics of fixed income.


Table 5.22 Matching a 15-stock portfolio to a portfolio equal-weighted in three industrialsectors. Stock positions chosen to eliminate factor exposures

Stock Factor Loadings $ Position $ Exposure Profit from factor move———————— —————————————— ————————————1 2 3 1 2 3 5.00% 1.00% �2.00%

1 0.78 0.12 �0.22 1,800,000 1,404,000 216,000 (396,000) 70,200 2,160 7,9202 0.81 0.07 �0.08 1,700,000 1,377,000 119,000 (136,000) 68,850 1,190 2,7203 1.17 0.02 0.06 1,600,000 1,872,000 32,000 96,000 93,600 320 (1,920)4 1.21 �0.04 0.20 1,330,000 1,609,300 (52,500) 266,000 80,465 (525) (5,320)5 1.25 �0.08 0.18 1,400,000 1,750,000 (112,000) 252,000 87,500 (1,120) (5,040)6 0.84 0.11 0.07 1,300,000 1,092,000 143,000 91,000 54,600 1,430 (1,820)7 0.87 0.04 �0.08 1,200,000 1,044,000 48,000 (96,000) 52,200 480 1,9208 0.90 �0.07 �0.15 1,150,000 1,035,000 (80,500) (172,500) 51,750 (805) 3,4509 1.07 �0.10 0.11 500,000 535,000 (50,000) 55,500 26,750 (500) (1,110)10 1.21 �0.17 0.18 900,000 1,089,000 (153,000) 162,000 54,450 (1,530) (3,240)11 1.34 0.33 �0.10 800,000 1,075,700 264,000 (80,000) 53,785 2,640 1,60012 1.49 0.11 �0.15 700,000 1,043,000 77,000 (105,000) 52,150 770 2,10013 0.93 �0.11 0.14 600,000 558,000 (66,000) 84,000 27,900 (660) (1,680)14 0.64 �0.33 0.11 500,000 320,000 (165,000) 55,000 16,000 (1,650) (1,100)15 0.99 �0.55 �0.19 400,000 396,000 (220,000) (76,000) 19,800 (2,200) 1,520

Sector 1 1.20 0.06 0.03 (5,000,000) (6,000,000) (300,000) (150,000) (300,000) (3,000) 3,000Sector 2 0.87 �0.05 �0.02 (5,800,000) (5,070,000) 300,000 116,000 (253,500) 3,000 (2,320)Sector 3 0.95 0.00 �0.01 (5,400,000) (5,130,000) — 34,000 (256,500) — (680)

Total — — — — (0) —

5.3 SPOT CURRENCY ARBITRAGE

Spot currency trading is dominated by speculation on the future direction of foreignexchange (FX) rates. Only a small component of trading is related to inventory adjust-ments of large wholesalers executing FX transactions on behalf of their correspondentbanks and retail customers. The speculation is mostly macroeconomics-driven both inthe short and long run. Bank economists and central bank watchers disseminatearriving economic data and their commentary to the trading desks, and currencytraders take bets on the effect of the information on FX rate movements.At the same time, an important activity called triangular arbitrage, introduced in

Chapter 2, takes place on the cross-currency desks. This activity ensures that FXrates, whether quoted relative to a vehicle or directly, stay in line with each other.Triangular arbitrage is an artifact of the essence of an FX quote as a price of moneyin terms of other money. With so many monies around, opportunities in the wholesalemarket arise when a currency may become relatively cheap when purchased with onecurrency, but not with another. Our description of the strategy will be static in that wewill assume that all quotes arrive at the same time. In reality, the arbitrage may be lessthan perfect as traders acquire one currency for another in anticipation of being able tosell it for another. Thus often there will be a time lapse between a purchase and a sale.The principle of arbitrage may be used to aid in setting up very short-timed speculativepositions. In our illustration of pure triangular arbitrage, we ignore transaction costsand assume that we can transact at mid-prices (i.e., there is no bid–ask spread). In theinterbank market, although the bid–ask spread is very small (of the order of the third


decimal place), it is not insignificant and it is the source of most of the trading profit forlarge institutions. It often renders a lot of triangular strategies unprofitable.

Suppose we observe that in New York the dollar/euro FX rate is [USD/EUR] 1.1235and the yen/dollar rate stands at [JPY/USD] 119.03. At the same time, in London theyen/euro cross rate is at [JPY/EUR] 132.85. How can we profit?

First, let us determine the yen price of the euro in New York. One euro costs 1.1235dollars and each dollar costs 119.03 yen. So, 1 euro can fetch 1.1235� 119.03 ¼ 133.73yen. This is more than the 132.85 that it can fetch in London. The same good, the euro,sells for two different prices. If we can buy and sell it in both places, we would like tobuy the euro in London and sell the euro in New York. Suppose we are endowed withEUR 1,000,000. We sell the euros in New York for dollars to get:

1,000,000½EUR� � 1:1235½USD/EUR� ¼ USD 1,123,500

We then sell the dollars for yen to get:

1,123,500½USD� � 119:03½JPY/USD� ¼ JPY 133,730,205

Lastly, in London we sell the yen for euros in the cross-market to get:

133,730,205[JPY]/132.85½JPY � EUR� ¼ EUR 1,006,626

and we end up with an instant profit of EUR 6,626. If we could, we would want to dothis transaction in much larger size; and everyone else would want to copy it.

The strategy is self-correcting. All three transactions in it act to move the FX rates inline with each other. The sale of euros for dollars will increase the supply of euros andtend to reduce the price of euros in dollars (e.g., to [USD/EUR] 1.1185). The sale ofdollars for yen will reduce the price of dollars in terms of yen (e.g., to 118.83). Overallthis will decrease the indirect value of the euro in New York to [JPY/EUR] 132.91. If, atthe same time, the yen value of the euro increases in the cross market in London as aresult of the third transaction (the purchase of euros), then the arbitrage opportunitywill disappear. In the dynamic world of currency trading, where FX rates change everyinstant, the opportunity of triangular arbitrage may depend on the speed of executionand a bit of luck.

The principle of triangular arbitrage is used by wholesalers in executing customercross-transactions and competing for retail business. Suppose a Polish trading companyis looking to exchange PLZ 5,000,000 into Swiss francs. The company obtains from amoney bank indirect quotes of [PLZ/EUR] 4.3545/4.3615 and [CHF/EUR] 1.1492/1.1505. In exchanging PLZ into CHF, the company would have to incur two bid–askspreads, resulting in the following effective market quotes: [PLZ/CHF] 3.7849/3.7952(bid for CHF: 4.3545/1.1505; offer of CHF: 4.3615/1.1492). By calling several moneybanks, the company can put them in competition to give up some of the profit. If abank were willing to do the more liquid CHF/EUR transaction at the mid of [CHF/EUR] 1.1499, then it could offer the Polish company a lower price for purchasing Swissfrancs of [PLZ/CHF] 3.7929. The wholesaler that maintains large inventories of CHFand EUR, but not PLZ, gives up the profit on the CHF/EUR leg, but still ends up witha profit on the PLZ/CHF leg of the trade. The customer gets a lower effective cost ofexchanging PLZ into CHF.


5.4 COMMODITY BASIS TRADES

Commodity basis trades are often entered into by non-financial institutions whosebusiness depends on commodity prices. These trades can be executed with the use offutures markets, but are governed by spot principles. The futures contract is used notbecause of the economics of the strategy, but mainly out of convenience and may be asource of additional risk. The trade is normally a hedge for revenues or costs. Largecommodity dealers execute similar strategies for speculative reasons.Consider a small experimental heating solutions company. Its heaters use specialized

fuel similar to heating oil, but more refined, to supply heat to customers at fixed pre-contracted rates. If the cost of the fuel it uses goes up, its profits will suffer. Thecompany wants to eliminate the price risk of the specialty fuel. However, the specializedfuel is not widely traded, but the standard #2 heating oil is.Let us assume that the price of the specialty fuel is equally volatile as that of the #2

heating oil. A 10% rise in the price of the heating oil is accompanied by a 10% rise inthe specialty fuel price. Suppose that, currently, heating oil trades spot at $0.85 percubic meter and the specialty fuel trades at $1.34. The company needs 50 million cubicmeters of the fuel. Let us also assume that there is an active heating oil futures marketand the price of a 6-month contract is equal to $0.05 more than the spot price, due tofinancing costs and storage conditions (i.e., $0.90). The size of the contract is 1 millioncubic meters. The company cannot use heating oil, but it can use the market for heatingoil as a hedge.The difference between the price of the specialty fuel and the heating oil, equal to 1.34

minus 0.85, or $0.49, is called a basis. The existence of a basis will lead to a hedge ratiothat is not one for one, similar to a hedge of two bonds with different durations. Let usdenote the heating oil futures contract size, or the multiplier, as m ¼ 1,000,000, theprice of the hedge instrument as H ¼ 0:85, the absolute basis as B ¼ 0:49, and thepercentage basis as b ¼ 0:49=0:85 ¼ 57:65%. In order to hedge its cost exposure,the company buys:

50 �HþB

H¼50 � ð1þbÞH

H¼50 � 1:34

0:85¼78:90

or, rounding, 79 heating oil contracts. Notice that on a per-contract basis, the hedgeratio will be more or less than unity depending on whether the basis b is positive ornegative.Suppose the price of specialty fuel increases by 15%, which translates into a potential

cost increase of 50,000,000 � 1:34 � ð0:15Þ ¼ $10,050,000, and that the percentage basisholds (i.e., the heating oil price increases also by 15% from $0.85 to $0.9775). Thefutures price, assuming a constant futures basis, rises from $0.90 to $1.0275. Thecompany’s gain on the long contracts is:

79(1,000,000)(1.0275� 0:9000) ¼ 10,072,500

The precise setup of the hedge depends on whether we believe that the relative orabsolute basis holds. It also depends on the assumed relationship of futures to spotfor the hedge instrument. The hedge is designed to work for any increase in the price ofthe underlying cost determinant.


Commodity trading houses use the concept of a basis to set up speculative trades withone commodity bought and another shorted. The basis is the hedge ratio used indetermining the relative proportions of the trade. In this application, the basis isused as a statistical concept, akin to the relationship of stock returns to factorsthrough loadings, rather than a mathematical one, akin to the duration in bonds.Current basis is compared with historical basis levels estimated using the pricehistory of the underlying and hedge instruments. It cannot be computed by discountedcash flow arguments.


___________________________________________________________________________________________________________________________________________ Part Two __________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________ Forwards __________________________________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________________________________________________________________ 6 __________________________________________________________________________________________________________________________________________________________________________

_________ Financial Math II—Futures and Forwards _________

The greatest misconception about the futures and forward markets is that they predictthe prices of commodities or other financial instruments in the future while spotmarkets, separately, establish their value now. In this view, corn can trade at oneprice today and at a totally different price for a 6-month delivery, presumably reflectingfuture supply conditions (e.g., poor crops due to a drought). While it is true that futuredemand and supply conditions enter the equation, what is important to note is that theyare joint determinants of both futures and spot prices. If everyone expects the droughtthat will drive up the price of the corn over the next 6 months, then why wouldn’t someenterprising farmers buy up the corn now and pay for storing it in a silo, to sell it at aprofit at a later date, resulting in a bid-up spot price today?The fact is that spot and forward prices are tied together through a cost-of-carry

relationship. The futures price must always be equal to the spot price grossed up by thecost of financing to purchase the commodity now and storing it, net of any cash flowsfrom holding the commodity. Thus, if there is an expectation of any impending supplyshocks, their impact is automatically translated not just into futures and forwards, butinto spot prices as well.The cost of carry is defined as the cost of financing and storing a commodity or a

financial instrument net of any cash flows accruing from the instrument between nowand some future date. In the spot–forward analysis, that future date is the delivery dateof the forward contract or the expiry date of the futures. A cash-and-carry transactionconsists of borrowing funds and purchasing an asset today, paying for storing it andpotentially receiving cash flows from it (dividends or coupons) between today and afuture date. A reverse cash-and-carry transaction consists of short selling an asset andlending the proceeds today, compensating the lender of the asset for storage costsand cash flows from the asset between today and a future date. The cash-and-carryargument is fundamental for establishing a fair value of futures and forwards.A forward contract is defined as a contract to buy and sell an asset at a price agreed

on today, but for a delivery at some future date. On the delivery date, the seller of theasset delivers the asset and receives cash for it. Today, no asset or cash changes hands.The two parties only agree on the sale price. The seller of the asset is said to go short,the buyer of the asset is said to go long on the forward contract. A futures contract isanalogous to a forward contract in that there is a buyer and a seller of some underlyingcommodity and that they agree on a future price (or other variable) today. Subsequentto that agreement, at the end of each day, they exchange cash flows equal to themovement in the price of that commodity (or in the movement of the variable) timesa multiplier. The process is referred to as marking-to-market; the settlement cash flowsare referred to as the variation margin. On the futures expiry date, the futures price of

the commodity is set equal to the current spot (by definition), so that the net cumulativevariation margin exchanged between the expiry date and the original date is equal to thedifference between that future spot price and the original agreed-on price. This elabo-rate mechanism allows farmers to lock in the sale price of a future crop. The farmer sellsfutures contracts at today’s futures price. By the time of the futures expiry, the farmerwill have received or paid the amount equal to the difference between today’s futuresprice and the then prevailing spot price. When he sells his crop spot, the marking-to-market cash flows compensate him for any difference between that spot and the originalfutures price, so that effectively he sells for the original futures price. This was theoriginal purpose of organizing the futures markets in Chicago in the mid-1800s.Since then, futures markets expanded geographically around the world and product-wise to cover non-commodity prices (like bonds and stock indices) as well as non-prices(like interest rates and atmospheric temperature readings).

Futures contracts on non-price variables, like temperature readings, are the bestexamples to study, because they expose the contracts for what they really are: betson future outcomes. In that aspect, they are very similar to horse racing and sportingevent bookmaking. In the latter, an underlying event’s outcome will be revealed on agiven future date. Today, buyers and sellers can agree through a trading mechanism ontoday’s mean expectation of the outcome. If more people believe that team A will winagainst team B they will push up the spread in team A’s favor until the spread reflectsthe mean number of points by which team A is expected to beat team B. As informationarrives in the market between now and the event date, the spread will vary reflecting thechanging expectation of the betting public. Sports betting does not provide for auto-matic marking to market. It is rather like a forward market with dealers willing to closeout bets prior to maturity at the current mark.

A futures contract on the temperature, like the one traded on the Chicago MercantileExchange (CME), is very similar to a soccer match bet. When the soccer match ends, itsoutcome is revealed. When the expiry month ends, the temperature readings in a givenlocation have been observed. Homes and apartments require heating when the tem-perature outside is low. A heating degree day (HDD) is defined as the number of days ina given month when the temperature falls below 65�F times the difference between 65�Fand the actual temperature reading on a given day. On the CME, contracts are tradedfor several U.S. cities and several winter months. Buyers of the futures contracts can betthat the actual HDDs in a particular month is above a certain number (the spread),while sellers believe that the actual number of heating days will be below that impliedby today’s spread. The contract has a multiplier of $100 per HDD which translates thedifferences between HDDs into dollars of payout. Suppose today two parties enter intoa 3-month expiry futures contract at a ‘‘price’’ of 52HDDs. One party goes long (buys)one contract, the other goes short (sells) one contract. Each day until expiry, the partieswill exchange cash flows equal to today’s closing futures price minus yesterday’s timesthe multiplier. For example, if the next day’s price is 56HDDs, then the long party willpay and the short party will receive ð56� 52Þ � $100 ¼ $400. They will continue ex-changing cash flows until the expiry date. On that date, the last cash flow will beexchanged equal to that day’s closing price, equal to the actual number of HDDs inthe expiry month on which they had bet in the first place minus the previous day’s pricetimes the multiplier. Because all the intermediate cash flows will wash, the total sum ofall the cash flows received by the long party and paid by the short party since inception


will be equal to the difference between the actual number of HDDs during the expirymonth and the original bet price of 52HDDs times the multiplier. If the result isnegative, then the long party will have paid and the short party received. Similarfutures contracts trade for all major U.S. cities based on cooling degree days (CDDs)defined analogously to HDDs.Stock index futures contracts work exactly the same way. Buyers bet that an index (a

number, not a price of anything) will go up and sellers that the index will go down. Thepayout is equal to the value of the index less the bet price, times a multiplier specified bythe exchange (e.g., $250 per point of the S&P 500 index per futures contract). Privateforward contracts analogous to futures contracts may be easily established betweentwo parties. There is no marking to market (intermediate cash flows) for forwards.Instead, there is only one settlement at the end, equal to the final price minus the bet(forward) price, times the notional amount of the contract (the multiplier). For mostliquid financial instruments, an intermediate settlement may occur if requested by oneparty.Apart from the marking to market which is automatic with futures and only if

mutually agreed to with forwards, there is one more fundamental difference betweenthe two vehicles: credit exposure. Futures are typically traded on an exchange thatoperates a clearinghouse. When two parties enter into a futures trade, even thoughthey must find each other on the floor and agree on the price, legally each transacts notwith the other, but with the clearinghouse that guarantees the payments. Neither partyis exposed to the other in case one defaults on its obligations. The clearinghouse’sguarantee is backed by a good faith futures margin each party and other partiesmaintain with it. The existence of a clearinghouse also enables parties to close outcontracts with counterparties other than the original one as each transaction is azero-sum game with the same number of longs as shorts and each is a legal transactionbetween a party and the clearinghouse. Futures contracts are standardized as tomaturity dates and contract amounts (multipliers). Forwards can be custom-tailored,but each party bears credit exposure to the other because they are private contractsbetween two parties.Futures exchanges in the U.S. typically have physical trading floors and the method

of trading is referred to as open outcry. The biggest electronic futures exchange isoperated by EUREX which after years of operating in Europe debuted successfullyin the U.S. in February of 2004 competing head to head with the Chicago Board ofTrade (CBT) in the U.S. bond contract. Both forms, physical or electronic, operate aclearinghouse that mitigates credit exposure and allows closing positions with anyparticipant. The only real difference is that on an electronic exchange the bids andoffers are displayed and transactions are entered into on computer screens ratherthan by people gesturing to each other.Despite being bets on future events, futures and forwards perform a very important

risk-sharing function. Parties unwilling to take the risk of future price or interest ratechanges or the risk of changing raw material costs due to temperature variations canlock in future spot prices at today’s futures and forward prices. They do not have towait to see what future spot prices will obtain. They can lock in their cost ofborrowing or lending, or the price they pay for coffee or oil. They cannot lockthat cost at any level, but only at that equal to today’s price in the forwards orfutures market.

Financial Math II—Futures and Forwards 137

In turn, the forward price is tied to the spot price through the cost of carry. Supposewe think of the S&P 500 index level as the price of a tiny basket of 500 stocks in theproportions represented in the index and that we want to lock in a price of acquiring250 of such baskets at a future date. We do not want to own the baskets today, but wewant to know for sure at what price we can buy them 3 months from today. By goinglong 1 futures contract (the multiplier is equal to $250 per index point), we are transfer-ring the price risk to the short party. If the index value is higher on the purchase date wewill have received cash flows from the futures contract, compensating us exactly for theprice increase. If the index value is lower, we will have paid into the futures contractgiving up any potential savings we could have gained. At what price are we going to beable to lock in the purchase price 3 months hence? The answer is price equal to today’sindex value adjusted for the cost of borrowing money between now and the futuresexpiry date. This follows from a simple arbitrage relationship. Instead of going long 1futures contract, we could borrow an amount of money equal to 250 times today’s valueof the index and purchase the baskets today. On the expiry date we have to return theborrowed cash, which is equivalent to a known cash outflow and a stock inflow, thesame as in the future purchase of the index. The cost of carry is thus the financing costof pre-purchasing the underlying commodity in the futures contract, net of any cashflows. In the case of the stock index, the extra cash flows are positive to us in the form ofintermediate dividends earned on the component stocks, which reduces our cost ofcarry (interest cost minus the cash flows).

6.1 COMMODITY FUTURES MECHANICS

Let us consider an example of a corn contract traded on the CBT. As reported in theWall Street Journal on Thursday, October 25, 2001, the December 2001 delivery cornsettled at 2061

2cents per bushel. Each contract represents a claim on 5,000 bushels of

corn.

Table 6.1 Grain and oilseeds futures prices for Thursday, October 25, 2001

Contract month Open High Low Settle Open interest

Corn (CBT) 5,000 bushels (cents per bushel):

Nov01 20134

20212

20112

202 2,429Dec01 206 2071

22051

22061

2211,016

Jan02 211 21214

21012

211 680Mar02 2183

42201

42181

4219 112,379

. . .

Soybean meal (CBT) 100 tons (in $ per ton):

Dec01 157.50 161.80 157.50 161.30 46,639Jan02 156.00 159.20 154.50 158.60 20,156Mar02 153.30 156.20 153.10 155.50 15,653

Source of data: Wall Street Journal, October 26, 2001.

The multiplier in the contract is $50 per 1 cent change in the futures price (5,000 bushels,but the price is in 0.01 of a dollar).


Suppose we buy one December contract (car) at the close price of 20612and observe

the following December settlement prices on the following two business days:

Friday, October 26, 2001 207.30Monday, October 29, 2001 206.80

Let us also assume that corn closed at 210.00 cents per bushel on the December expirydate (third Friday in December) which was also the spot price for corn on that day.This equivalence is assured because any outstanding long contract at expiry is a con-tract to purchase corn with a spot delivery from the short party.Here is how marking to market works. At the time we ‘‘bought’’ the contract, our

broker on the exchange floor found a seller and agreed to the transaction. We did notreceive any corn and we did not pay cash for any corn. We merely agreed on the size ofthe transaction (1 car) and the price (206.5). Since we bought the contract at the close,after the close on Thursday we did not pay or receive any cash flows (if the contract isbought in the middle of the day and the close price is different from the transactionprice, marking to market starts the same day). We bought one contract and the pricegoes up on Friday. On Friday after the close of the market, we receive from theclearinghouse a variation margin check for:

ð207:30� 206:50Þ � 50 ¼ $40

The clearinghouse has received a check for $40 from a short party. On Monday, theprice goes down, so we compute our cash flow as:

ð206:80� 207:30Þ � 50 ¼ $� 25

After the close of the market we send a check to the clearinghouse for $25 (which isforwarded to a short). Our net variation margin after 2 days is equal to $15 (¼ 40� 25).Notice that this amount is also equal to the close price on Monday minus the originalfutures purchase price times the multiplier:

ð206:80� 206:5Þ � 50 ¼ $15

as Friday’s close price of 207.30 washes out. It will be true for any day until expiry thatthe net variation margin, or the cumulative total sum of all the cash flows, will be equalto that day’s close price minus the original price times the multiplier. On the expirydate, with the corn price at 210.00, our net variation margin will be equal to:

ð210:00� 206:5Þ � 50 ¼ $175

Suppose we are a popcorn maker and the original purpose of entering into the longcorn contract was for us to lock in a purchase price of 2061

2cents per bushel for 5,000

bushels of corn, our main raw material. Did we accomplish that? On the expiry date wepurchase corn in the spot market for 210 cents per bushel. The total we pay is:

5,000� 2:10 ¼ $10,500

At the same time we have collected $175 from our hedge strategy, so that our net costwas $10,325. That cost translates into:

$10,325 net total/5,000 bushels ¼ 2:065 ¼ 206 12per bushel


Had the price of corn gone to 200 cents per bushel by December, we would have stilllocked in our price of 20612 since we would have collected from the long futures contract:

ð200:00� 206:50Þ � 50 ¼ $� 325

so that our cost of acquiring corn spot inclusive of the futures loss would have been thesame:

5,000� 2:00þ 325 ¼ $10,325

(i.e., 20612cents per bushel). In fact, no matter at what price corn would end up in

December, our cost would be 20612cents. We would be happy with the hedge if the spot

price in December is higher than the original futures price (as we would save). Wewould be unhappy if the spot price of corn in December is lower, as we would haveforegone a potential savings. Hedging in futures can be viewed as a two-way insurancestrategy. If the price moves as you anticipated (relative to the lock-in value), you getreimbursed; if the price moves in the opposite direction, you reimburse someone else.You lock in a cash flow, but you do not eliminate the possibility of a mark-to-marketloss. In that sense, locking future cash flows can be viewed as speculation.

Let us test our understanding of the mechanics of commodity futures by answeringthe following questions related to the same excerpt from the Wall Street Journal:

Q1. What is the multiplier for soybean meal?Q2. If you sell 5 cars of Mar02 soybean meal futures at 155.50 and tomorrow the price

goes to 158, what is your cash flow (variation margin)?Q3. If at expiry in March, soybean meal trades at 160, what is your net variation

margin?Q4. What sale price did you lock in on the sale of 500 tons?

Here are the answers:

A1. Each contract is for 100 tons of soybean meal and prices are in $/ton, so themultiplier is 100.

A2. If we shorted 5 contracts at the close on Thursday at 155.50 and the next day theprice went down, then we lost money (i.e., had a negative cash flow, or variationmargin) for that day of:

ð�5Þ � 158� 155:50Þ � 100 ¼ $� 1,250

A3. If the price at expiry is 160, then between October and March we have accumu-lated a net variation margin of:

ð�5Þ � ð160� 155:50Þ � 100 ¼ $� 2,250

A4. If we sell 500 tons of soybean meal at $160/ton for $80,000 and we lost $2,250 onfutures, then the total price per ton we locked in is:

ð80,000� 2,250Þ=500 ¼ $155:50

The word multiplier is not commonly used with commodity futures. Instead, we refer tothe contract size (i.e., the amount of commodity represented in each contract). Contractsize can always be translated into a multiplier. We use the latter exclusively to make theunderstanding of the next sections easier and to emphasize that futures contracts can bewritten on any variable (e.g., index, temperature, interest rate) whose changes have to


be translated into dollars to settle the variation margin every day. With non-pricevariables, the multiplier is explicitly defined.

6.2 INTEREST-RATE FUTURES AND FORWARDS

Overview

Most interest rate-related futures contracts around the world are defined based onactual or artificial bond prices, not directly on interest rates. For example, on theLIFFE in London, the long gilt contract is defined on GBP 500,000 face value of thelong bond of the U.K. government. On the EUREX, one 10-year Euro-Bund futurescontract represents EUR 100,000 of the euro-denominated bond of the German gov-ernment. Like spot, bond futures prices are quoted as a percentage of par value. InChicago, on the CBT, one Treasury Bond contract represents USD 100,000 face valueof a 30-year Treasury bond. The futures price is quoted as a percentage of par value(with fractions in 32nds). To compute the variation margin for a given day, we firsthave to translate it to a straight percentage and then multiply by the size of the contract.For example, if we shorted five contracts at 112-03 and the price changed to 112-27,then we would have a loss, or a negative variation margin amount, of:

ð�5Þ�ð112 2732�112 3

32Þ% of 100,000 ¼ ð�5Þ � ð0:75=100Þ�100,000 ¼ $�3,750

Note that the multiplier, as defined before, is in this case equal to 1,000. A 0.75 changein the price represents $750 variation margin per contract. If we wanted to speculatedirectly on a specific size of an interest rate change and not a bond price change, wewould have to scale our bet based on the current duration of the underlying bond. Tobet on an interest rate increase of 1% for $1,000,000 face value of bonds, one wouldhave to short [(1/Duration)� 10] bond contracts. As interest rates increase by 1% thevalue of each of the 10 contracts representing $100,000 face value, for a total of$1,000,000 par value, would decrease by 1 times the duration number of points, pro-ducing the desired dollar gain.All U.S. Treasury bond and note contracts, and many other government bond

contracts, are defined not on one underlying bond, but on a set satisfying certainmaturity and coupon criteria (e.g., close in maturity and coupon to the 30-year 6%mark). The short party to the contract is given an option to choose which of the eligiblebonds to deliver to the long party on the futures expiry date. The short party is alsogiven additional timing options that complicate the analysis of these contracts. Thereason behind this complication is to ensure the liquidity of the delivery instrumentswhich are considered fungible.Besides the government bond futures for the benchmark issues in each market (U.S.,

U.K., Bunds in Germany, JGBs in Japan), the most popular futures contracts are thosewritten on Eurocurrency deposits. The contract with the most turnover in the world isthat on the 3-month Eurodollar deposit rate (i.e., the 3-month USD LIBOR). It istraded on the CME and is fungible with the same contract traded on the SIMEX inSingapore. Similarly, contracts are traded on yen- and euro-denominated Eurodepositsas well as other Eurocurrencies. Next we review the somewhat peculiar mechanics ofEurocurrency futures.


Eurocurrency deposits

Recall from Chapter 3 that a Eurodollar deposit (a ‘‘depo’’) is a non-negotiable, U.S.dollar-denominated, interest-bearing loan on deposit outside of the U.S. regulators’reach. The interest rate is fixed and quoted on an Act/360 basis and paid as add-oninterest at the end. There is a variety of terms available ranging from overnight,tomorrow/next day, 1 day, and so on, all the way to 12 months. The most popularone is a $1,000,000 3-month deposit with a 2-London-business-day settlement. Euro-dollar lending rates are fixed daily by the group of London banks most active in thismarket in the form of a London interbank offered rate (LIBOR), widely published byall financial services and newspapers; the bid on borrowed funds is referred to asLIBID, but rarely quoted. The LIBOR rate is the benchmark rate for unrestricted 3-month deposits and swap payments’ settlements. The Eurodollar market is the largestglobal money market by volume outstanding and turnover. The interest mechanicswork as follows. Suppose on Thursday, October 25, 2001 you called a bank toborrow Eurodollars at the interest rate of 2.31% for a term of 3 months. Yourinterest accrual period ran from Monday, October 29, 2001, to Tuesday, January 29,2002. This reflects the customary 2 business day settlement period. Assuming there were92 days in the period, your payment of interest and principal on January 29, 2002 wasequal to:

1,000,000� ð1þ 0:231� 92=360Þ ¼ $1,005,903:33

In the following, we will ignore the 2-day settlement delay. We will pretend that a 3-month deposit arranged today starts today and ends 3 months later. Actually, it wouldstart 2 days after today and end 3 months after that. We will also assume that 3 monthsequals exactly 90 days.

Eurodollar futures

The Chicago Mercantile Exchange (CME) Eurodollar contract was designed to allowlocking in the rate paid or received on a $1,000,000 3-month Eurodollar (ED) depositstarting on the futures expiry date and ending 90 days later. To that end, it is quoted onan artificial price basis. The futures ‘‘price’’ is defined only for the expiry date as:

F ¼ 100� L

where L is the 3-month LIBOR rate. Note that on the expiry date, F is not the price of a$1,000,000 deposit or even the price of a discount instrument promising to pay 100 at Lpercent interest 3 months later. Instead, on the expiry date, L ¼ 100� F . Every day wecan interpret 100� F as the LIBOR rate L that can be locked in on that day for aforward deposit starting on the futures expiry date. For example, a price of 97.69 meansthat a rate of 2.31% can be locked in. The stated size of the contract of $1,000,000 isalso somewhat misleading unless we know from the fine print that the multiplier is$2,500 per point or $25 per basis point. Let us consider some real quotes as of theclose of day on Thursday, October 25, 2001.


Table 6.2 Interest rate futures prices for Thursday, October 25, 2001

Contract month Open High Low Settle Open interest

Eurodollar (CME) $1 million (in points of 100%)

Nov01 97.79 97.83 97.78 97.81 41,554Dec01 97.81 97.87 97.80 97.85 836,180Jan02 97.86 97.88 97.86 97.87 7,080Mar02 97.71 97.80 97.71 97.77 628,766Jun02 97.41 97.51 97.41 97.48 588,920Sep02 96.98 97.12 96.98 97.09 398,280Dec02 96.50 96.63 96.47 96.60 378,297Mar03 96.14 96.25 96.14 96.24 243,043. . .

Euroyen (CME) Y¼100 million (in points of 100%)Dec01 99.91 99.91 99.91 99.91 19,080. . .

Source of data: Wall Street Journal, October 26, 2001.

The June 2002 contract closed at 97.48. That contract covers a forward deposit startingon June 19, 2002 and ending on September 19, 2002. The implied forward rate for thatdeposit is 100� 97:48 ¼ 2:52%. The spot LIBOR rate on October 25, 2001 was 2.31%.For simplicity let us assume that there are exactly 90 days in the June 19–September 19period.Suppose at the close on Thursday, we go long on one ED contract and over the

following two business days we observe these futures prices:

Friday, October 26, 2001 97.44Monday, October 29, 2001 97.55

Our variation margin on one long contract would be:

October 26, 2001 ð1Þ � ð9744� 9748Þ � 25 ¼ $� 100October 29, 2001 ð1Þ � ð9755� 9744Þ � 25 ¼ $275

We dropped the decimals to convert from points (bp) to basis points and then used themultiplier in $/bp. Our net variation margin as of Monday would be $175 which alsocould be computed simply from the Monday close price and the original price as:

ð1Þ � ð9,755� 9,748Þ � 25 ¼ $175

Locking in a deposit rate

We consider two scenarios for LIBOR as of June 19, 2002.

LIBOR ¼ 3% First suppose that on June 19, 2002 the 3-month LIBOR rate is3.00%. That automatically means that the last futures price is F ¼ 100� L ¼100:00� 3:00 ¼ 97:00, and our net variation margin over the October–June period is:

ð1Þ � ð9,700� 9,748Þ � 25 ¼ $� 1; 200


Suppose the original reason that we had entered into the long contract was to lock inthe rate of 2.52% we would earn on a $1,000,000 90-day deposit, and as planned onJune 19 we deposit $1,000,000 at 3.00%. Our interest on that deposit will be:

1; 000; 000� 0:03� 90=360 ¼ $7; 500

Including our loss of $1,200 on the futures contract, our effective interest rate earned is:

ð7,500� 1,200Þ=1,000; 000� ð360=90Þ ¼ ð6,300Þ=1,000,000 � ð360=90Þ ¼ 2:52%

LIBOR ¼ 2% If the LIBOR rate on June 19 is 2.00%, then the last futures price onthat day is F ¼ 100� L ¼ 100:00� 2:00 ¼ 98:00, and our net variation margin over theOctober–June period is:

ð1Þ � ð9,800� 9,748Þ � 25 ¼ $1,300

If on June 19 as intended we deposit $1,000,000 at 2.00%, our interest on that depositwill be:

1,000,000� 0:02� 90=360 ¼ $5,000

Including our gain of $1,300 on the futures contract, our effective interest earned isagain:

ð5,000þ 1,300Þ=1,000,000x� ð360=90Þ ¼ ð6,300Þ=1,000,000 � ð360=90Þ ¼ 2:52%

In fact, no matter what LIBOR is on June 19, our effective interest rate earned inclusiveof the futures gain/loss is $6,300, or 2.52%. If we had wanted to lock in the rate on a$25 million deposit, we would have had to buy 25 contracts. Next let us look at anexample of a borrower of Eurodollars.

Locking in a borrowing rate

On October 25, 2001 you want to lock in an interest rate on a $20 million loan startingon June 19, 2002 and ending on September 19, 2002. You can lock in a rate of 2.52%(but not any other rate) by shorting 20 ED contracts and waiting to borrow spot onJune 19.

LIBOR ¼ 3% If on June 19 the 3-month LIBOR is at 3.00% and the futures price isat 97.00, then you have a positive total cash flow from your futures contracts:

ð�20Þ � ð9,700� 9,748Þ � 25 ¼ $24,000

You borrow $20 million at 3.00% to incur interest cost on the loan of:

20,000,000� 0:03� ð90=360Þ ¼ $150,000

Including your futures gain your effective borrowing rate is:

ð150,000þ24,000Þ=20,000,000�ð360=90Þ¼ð126,000Þ=1,000,000�ð360=90Þ¼2:52%

LIBOR ¼ 2% If, instead, on June 19 LIBOR is at 2.00% and the futures price is at98.00, then you have a negative total cash flow from your futures contracts:

ð�20Þ � ð9,800� 9,748Þ � 25 ¼ $26,000


You borrow $20 million at 2.00% to incur interest cost on the loan of:

20,000,000� 0:02� ð90=360Þ ¼ $100,000

Including your futures gain your effective borrowing rate is:

ð100,000þ26,000Þ=20,000,000�ð360=90Þ¼ð126,000Þ=1,000,000�ð360=90Þ¼2:52%

So, we lock in a rate of 2.52% no matter where Libor ends up on June 19. Let usprovide another application of Eurodollar contracts, that of a lending or borrowingextension at a guaranteed rate through a rollover or bundling strategy.

Loan extension

Suppose a London bank quotes a rate of 2.38% on deposits for the 237-day periodstarting on October 25, 2001 through June 19, 2002. You have excess funds of $600million that you want to deposit through December 19, 2002. Using the ED futurescontracts, your strategy is the following:

. Deposit $600 million spot @ 2.38%.

. Do long Jun02 ED futures @ 97.48 and Sep02 ED futures @ 97.09 today to lock inreinvestment rates.

. As the spot deposit matures in June, reinvest in a 3-month deposit at the then-prevailing spot LIBOR rate.

. As the June deposit matures in September, reinvest in a 3-month deposit at the then-prevailing spot LIBOR rate.

We need to compute the number of June and September futures contracts we are goingto enter into. Our spot deposit will by June 19 accrue to:

$600,000,000½1þ 0:0238ð237=360Þ� ¼ $609,401,000

So, we need to guarantee a reinvestment rate for the June–September period for thatamount. We do that by going long 609 June02 ED contracts at 97.48. This ensures thatincluding any gain or loss on the futures, we will reinvest at 2.52%. By September 19,2002 our reinvested deposit will accrue to:

$609,401,000½1þ 0:0252ð92=360Þ� ¼ $613,325,542

Now, to lock in the rate of 2.91%, inclusive of any futures gain or loss, for theSeptember–December period, we will go long 613 Sep02 ED contracts at 97.09. Thiswill ensure that by December 19 we can expect to withdraw from our deposit account:

613,325,542½1þ 0:0291ð91=360Þ� ¼ $617,837,063

We have effectively locked in a rate of (17,837,063/600,000,000)(360/420) ¼ 2.5482%on an Act/360 basis for the entire period from October 25, 2001 through December 19,2002. To lock in a borrowing rate we would have computed the same amounts, but wewould have had to short futures. Also notice that in our strategy we assumed that weroll the interest over in the way money market accounts do when investors selectreinvestment of interest and dividends. Had we chosen to ‘‘consume’’ accrued interestin the form of cash withdrawals along the way, the number of futures contracts to beshorted would have been a constant 600.


It is easy to show that no matter what future spot LIBORs are, the rollover strategyproduces the desired result subject to a small error due to the fact that we cannot buyfractional ED contracts as required by our calculations.

Certainty equivalence of ED futures

It is important to notice one fundamental relationship governing the Eurocurrencyfutures contracts. An investor expecting to have excess funds available for deposit onJune 19, 2002 has two choices: to wait till June and invest the funds at the then-prevailing LIBOR rate (i.e., to bear the risk that the 3-month deposit rate willchange between now and then) or to lock in the rate by buying ED futures. Hecannot choose the rate that he can lock in; a seller has to agree to it too. (The rate isthus determined in the ED futures market.) It costs the investor nothing to enter into thefutures contracts. He merely needs to agree on the number of contracts and the pricewith a futures seller.

Instead of buying futures, the investor could call a bank and ask the bank to quote arate on a forward 3-month deposit starting on June 19. The bank would have to quote2.52%. If the bank were to quote more, the investor could arbitrage the bank byagreeing to the banks higher rate and shorting futures. This would earn him certainpositive cash flows on June 19. If the bank were to quote less than 2.52%, the investorwould go to the futures market to buy contracts and could again try to arbitrage thebank by trying to borrow from the bank at the quoted rate. The important observationhere is that the bank and the investor must be indifferent between the uncertain futurespot rate or today’s known rate implied in the futures price, as it costs nothing toconvert one into the other. This relationship will prove of paramount importancewhen valuing forward rate agreements and swap contracts whose one leg contains astring of LIBOR-dependent cash flows. Although the future LIBORs are unknowntoday, the present value of these cash flows will be computed by replacing theunknown future LIBORs by today’s forward rates or those implied in today’sfutures contracts. We can say that the market places a value on the unknown June19 LIBOR. It equates it to a known rate of 2.52%. That rate is not determined in thespot market, but by the demand for and supply of deposits in a separate market fordelivery of funds in the future.

Let us show one example of arbitrage forcing the rate of 2.52% to prevail in the bankmarket. Suppose we find a bank that offers a rate of 2.64% on June–Septemberdeposits. We agree to deposit $1,000,000 with a delivery of June 19. At the sametime, we short one ED contract to guarantee a borrowing rate of 2.52%. On June19, we borrow $1,000,000 in the spot deposit market at the then-prevailing LIBORand we deposit that $1,000,000 at our bank at a rate of 2.64%. We know that no matterwhat LIBOR is on June 19, our cost of borrowing is guaranteed to be 2.52%, assumingfor simplicity exactly 90 days. If LIBOR is 2.00%, we lose $1,300 on futures and pay$5,000 in interest, making it a total of $6,300. If LIBOR is 3.00%, we gain $1,200 onfutures, but pay $7,500 in interest, again making it a total of $6,300, which is equivalentto 2.52%. The bank guarantees to pay us:

1,000,000 ½1þ 0:0264ð90=360Þ� ¼ $1,006,600

So we earn $6,600 and we pay only $6,300, ensuring a $300 profit.


Borrowers and lenders in the Eurodollar market can share the risk of future interestrates with other investors by locking their funding rates in the futures and forwardmarkets. If they desire to lock in rates different from those quoted in the futures, thenthey have to compensate or are compensated for the present value of the differencebetween their desired rate and the certainty equivalents quoted in those markets timesthe appropriate day-count fraction.

Forward-rate agreements (FRAs)

In the over-the-counter (OTC) forward markets, the equivalent of the Eurocurrencyfutures contract is a forward-rate agreement (FRA). FRAs (pronounced ‘‘frahs’’) aremore flexible than futures in that they can be entered into for any future dates and forany notional principal amount. The disadvantage is there is no credit risk mitigation bya clearinghouse, as they are private contracts between two private parties. In order tounwind them, the two original parties have to agree on the mark-to-market value.Many of these issues are addressed by the use of contracts standardized by an inter-national industry association. These provide for netting for credit purposes (cashinflows and outflows for all contracts with the same counterparty are netted in caseof default and only the net amount is exposed to default) and using reference dealers incase the two parties cannot agree on the mark-to-market value for the unwinds ofprevious transactions. FRAs can be arranged by mutual agreement for any futuredates. Standard maturities are quoted daily on financial screens by larger dealers.These are listed for different start and end date combinations, relative to today,using the convention of ‘‘start month� end month’’. For example, in the quotes:

1� 4 2.75/2.762� 5 2.82/2.833� 6 2.84/2.85

the ‘‘2� 5’’ contract is bid at 2.82 and offered at 2.83. In this case, 2� 5 means acontract with a start date of exactly 2 months from today and the end date ofexactly 5 months from today (plus the number of days to settlement1). Just like thefutures contract, the FRA allows us to lock in a borrowing or lending rate for a future3-month period for indicated dates. The convention is for the dates to be definedrelative to today’s date. In the futures markets, these are defined absolutely in termsof actual calendar dates. Other standard forward periods are quoted in the market, for1-, 6-, and 12-month intervals (e.g., 2� 3 or 1� 7), following deposit maturity dates inthe Eurodollar, Euroeuro, or Euroyen markets. The language convention in the FRAmarket differs from that in the futures in that a dealer does not go long or short, butquotes the rates he is willing to ‘‘pay’’ or ‘‘receive’’ on the FRA. This is similar to theconvention used in interest rate swap markets.Let us describe the exact mechanics of a FRA contract. A FRA is equivalent to a

cash-settled fixed-for-floating swaplet (i.e., a swap with only one rate set and only one


1 This is governed by the FRABBA rules of the British Bankers Association (BBA) for FRAs. These are identical to those forLIBOR deposits and swaps. For example, for U.S. dollars, FRA spot is 2 days after the trade date and LIBOR is set 2 daysprior to spot and roll dates (for swaps). For pound sterling, spot is same date with same date LIBOR sets. The payout is madeon the settlement date at the start of the FRA contract period.

exchange of cash flows on the pay date; see Chapter 8 for swap definitions). One partyagrees to pay and the other agrees to receive a fixed rate of interest applied to a notionalprincipal amount over a 3-month period in exchange for receiving (the other partypaying) a floating rate equal to the spot 3-month LIBOR rate on the forward startdate, applied to the same principal amount. The pay and receive amounts are netted,present-valued by the spot LIBOR on the forward start date, and settled in cash on thatdate. Suppose on March 19, 2002 we entered into a 3� 6 FRA to pay 2.52% on$20,000,000. The start date of the forward is June 19 and the end date September 19.The amount the ‘‘FRA payer’’ will pay to the ‘‘FRA receiver’’ is defined as:

20,000,000� ½ð2:52� LÞ=100� � ðAct=360Þ � ½1=ð1þ L� Act=36,000Þ�The payment has the difference between the agreed-on rate and LIBOR applied to thenotional principal, scaled by the appropriate day-count fraction, and multiplied bythe present value factor for 3 months. The payment is computed on June 19, whenthe LIBOR rate L is revealed, and remitted 2 business days later on June 21. The cashflow can be positive or negative depending on the LIBOR rate on June 19 relative to theupfront forward rate.

Suppose both parties wait till June 19 to observe the then-prevailing spot 3-monthLIBOR to turn out to be 3.00%. The settlement cash flow is then computed as:

20,000,000� ½ð2:52� 3:00Þ=100� � ð92=360Þ � ½1=ð1þ 0:03� 92=360Þ� ¼ $� 24,346:68

The FRA payer would thus receive a check for $24,346.68 from the FRA receiver. Thisis almost exactly the $24,000 that the seller of 20 ED futures contracts would receive inthe form of net variation margin by June 19. The difference is that the FRA cash flow is(1) computed for 92 days and not for 90, as with a $25 multiplier, (2) it is present-valuedby 3 months (from the end date to the start date), and (3) it is received all at once onJune 19, instead of over time in the form of daily variation margin checks. For shortforward start dates, the difference between forwards and futures is negligible. Forlonger forward start dates, 2 years and beyond, the difference grows and the tworates start diverging. The difference between the quoted FRA and futures rates forthe same dates is referred to as futures convexity and is largely due to the timingmismatch of the cash flows causing futures to be price–yield convex instruments relativeto forwards. This is analogous to viewing bonds as having a convex price–yield relation-ship and forwards as having a straight-line relationship to forward yields. The futuresconvexity charge can be quite large (in the order of several tens of basis points for 5- to7-year futures) if the interest rate volatility between now and the expiry date is high.

It is easy to compute in our example that if the LIBOR rate ends up at 2.00 on June19, then the FRA payer will have to send a check to the FRA receiver for the amountof:

20,000,000� ½ð2:52� 2:00Þ=100� � ð92=360Þ � ½1=ð1þ 0:02� 92=360Þ� ¼ 26,442:63

which is close to the $26,000 we computed in the ED futures example, except for a smalldifference due to the same factors. We can conclude that, just like the ED futurescontract, the FRA allows us to lock in a net borrowing/lending rate for a futureperiod by simply matching the principal amount and choosing the correct side of thecontract. The FRA payer locks in a borrowing rate, while the FRA receiver locks in thelending or deposit rate. The easiest way to remember which side is which is to think of


borrowing as issuing a bond and paying a fixed rate on it, hence locking in a borrowingrate requires paying on the FRA or selling futures. The latter can be thought of asselling a forward discount bond. We can think of lending as buying a bond and receivinga fixed rate on it, hence locking in a lending rate requires receiving on the FRA orbuying futures. The latter can be thought of as buying a forward discount bond.

Certainty equivalence of FRAs

We use the term certainty equivalence to mean that the market determines today a fixedrate at which it is indifferent to exchange (i.e., willing to exchange at no charge) futureknown cash flows for future unknown cash flows based on future spot deposit rates.The certainty equivalence of FRAs is even more obvious than that of the ED futures,simply by looking at the settlement formula. The formula reflects the exchange of theforward rate known today for a future LIBOR rate, not known until the start of theforward interest accrual period. The exchange of the rates is applied to the samenotional principal and scaled by the appropriate day-count fraction. It is present-valued by 3 months to correct for the fact that, normally, interest is computed at thebeginning of the accrual period, but paid at the end, whereas the FRA settles at thebeginning of the period. The correction is exact, not approximate, since we applythe correct 3-month LIBOR rate as of the beginning of the accrual period (i.e., therate at which we could borrow/lend against cash flows occurring 3 months later). Thecertainty equivalence of future spot LIBOR to today’s forward rate comes from the factthat the FRA settlement formula is agreed on today, but with no cash changing handstoday. That is, the two parties to an FRA are indifferent between the two rates.We will come back to the certainty equivalence of futures and forwards later in this

chapter when we discuss their relationship to spot zero instruments.

6.3 STOCK INDEX FUTURES

There are futures contracts traded on all major stock indexes around the world. InChicago and New York, futures contracts are traded on a variety of U.S. stock indexesranging from the most popular S&P 500 to the broad Russell 2000. The CME tradesalso a Nikkei 225 contract. In Europe, there are futures on the FTSE 100 on LIFFE,CAC-40 on MATIF, DAX on EUREX, and a variety of others.All stock index futures are defined with the use of a multiplier that converts the

points of the index into a currency of denomination. The CAC-40 futures onMATIF are defined as c¼ 10 times the index. The FTSE 100 contracts on LIFFE are£10 times the index. The Mini-NASDAQ 100 contract on the CME is $20 times theindex. An interesting one is the Nikkei 225 on the CME which is defined as $5 times theindex. While the other examples convert an index into its ‘‘home’’ currency, the Nikkeicontract converts it into a ‘‘foreign’’ currency. This is not as peculiar as we might think.The outcome of a Real Madrid vs. Juventus soccer match is converted into Britishpounds by a London bookmaker and into euros by an Irish one. The outcome of asporting event cannot be traded in the spot market, neither can a stock index. However,financial dealers can buy and sell baskets of stocks in their home currency in the exactproportions of the index, creating synthetic spot trading in the index (they cannot buy


the Nikkei stocks for dollars). This has implications for the spot–futures price relation-ship through the cost of carry.

On July 22, 2003 the Wall Street Journal reported the following futures settlementprices for the S&P 500 index equal to 978.80 (spot) as of the close of the previous day,Monday, July 21, 2003:

S&P 500 Index (CME)—$250� indexSep03 978.00Dec03 976.10Mar04 974.20

On that same day, the U.S. dollar LIBOR rates stood at:

1-month 1.100 0%3-month 1.110 0%6-month 1.120 0%12-month 1.211 25%

To simplify the analysis, we will use a continuously compounded rate of 1.13% for allexpiry dates (1.1293% is the continuously compounded equivalent of the 6-month rateof 1.12% on an Act/360 basis assuming 182 days).

Locking in a forward price of the index

The index futures market allows investors to lock in a price at which they can buy or sella basket of stocks represented in the index. Today’s value of an index of 978.80 can bethought of as a $978.80 price of a basket of 500 stocks bought in very small quantitiesbut in the right proportions. By going long one December futures contract we can lockin today the price of acquiring 250 such baskets on the expiry date in December (usingthe multiplier of $250 per index point). The argument here is similar to that used abovefor soybean meal. The net variation margin will compensate us for any loss or takeaway any gain we may face as a result of the price difference between the future spotprice and today’s futures price.

Suppose we go long one December contract at 976.10 and on the December expirydate the spot index value is 1,000.00. Our net variation margin will be an inflow of:

250� ð1,000:00� 976:10Þ ¼ $5,975

If in December we buy 250 baskets at $1,000 per basket we will pay $250,000, whichcombined with a savings of $5,975 will yield a price per basket of:

ð250,000� 5,975Þ=250 ¼ $976:10

Fair value of futures

Let us now review the theory of futures pricing based on arbitrage, which can be foundin finance textbooks. This states that, for an asset that can be stored at no cost and


which does not yield any cash flows, the futures price F must be equal to the spot priceS plus the cost of financing the purchase of the underlying asset spot between the spotdate and the expiry date; that is:

F ¼ S þ ðFinancingÞ

Equivalently, this is equal to the spot price that is future-valued to the expiry date. Thistheoretical futures price is commonly referred to as fair value. For an asset whosepurchase can be financed till expiry with a loan rate of LIBOR L, the definitiontranslates into the following intra-year equation:

F ¼ S

�1þ L� Act

360

�

Note that the right-hand side has two components: S, the spot price, and SL� Act

360, the

interest cost on borrowing the amount S till expiry. Using an equivalent continuouslycompounded rate r, the equation is often written as:

F ¼ Sert

where t is the time to maturity in years (e.g., t ¼ 12for 6 months). Recall from Chapter 2

that when using continuous rates the discrete rate ð1þ rÞn expressions get replaced byert). The theoretical futures price expression, which simply reflects the cost of carry, isguaranteed by an arbitrage or a synthetic replication argument.Let us numerically illustrate the replication argument. Suppose today’s value of an

index is 978.80; this can be thought of as a $978.80 price of a basket of 500 stocksbought in the right proportions. We want to replicate exactly the cash flows of onefutures contract with a multiplier of $250 per index point and expiry of 3 months. Thecontinuously compounded rate at which we can borrow or lend funds is equal to1.13%. Instead of going long one futures contract, we can do the following:

. We borrow 978.80� 250 ¼ $244,700 at a rate of 1.13%.

. We purchase 250 baskets of stocks in the index (i.e., 250 times the numbers of sharesrepresented in one basket) at a cost of 978.80� 250 ¼ $244,700.

This results in a zero net cash flow today. On the futures expiry date:

. We pay off the principal and interest on the loan with a cash outflow of:

250� Se0:0113�ð1=4Þ ¼ 244,700e0:0113�ð1=4Þ ¼ $245,392:25

. We sell the stock baskets at the spot index price per basket St on the expiry date andreceive cash for them.

Our net cash flow on the expiry date is: 250� St � $245,392:25.Assuming that the entire net variation margin is received/paid on the expiry date,

going long one futures contract can be seen to be equivalent to the above strategy, inthat it results in a net cash inflow equal to 250� St, unknown today, minus 250 timessome amount of dollar amount, equal to the original futures price. If there exists afutures market, then traders will have a choice of engaging in the above replicating


strategy or going long futures. The two options must yield the same cash flows at alldates (spot and future), otherwise people would choose only the one with a higher netcash inflow. By equating the two, we get that:

250� St � $245,392:25 ¼ 250� ðSt � FÞ

The left-hand side is the cash flow for the replicating strategy on the expiry date, andthe right-hand side is the definition of the net variation margin. This yields us theposited futures–spot equation and the futures price of:

F ¼ 978:80e0:013�ð1=4Þ ¼ 981:57

Thus we have shown that a long futures contract can be replicated through a cash-and-carry transaction. That is, a buyer of futures, in effect, does not need the futures market;he can accomplish the same by buying the underlying asset in the spot market,combined with a credit market transaction (borrowing). In order for the fair valuerelationship to hold, we must show that the seller of futures does not need thefutures market either, in that he can guarantee the same spot and future cash flowsas the redundant futures contract by short selling the underlying asset spot and lendingin a credit market. This is called a reverse cash-and-carry strategy. For pure assets likestocks, this is quite easy. The amounts are the same; only the direction of the transac-tions is the opposite (short the asset and lend to earn interest).

Instead of the replication argument, we can also use a direct arbitrage argument byconsidering and refuting two alternative cases: (1) F > 981:57 and (2) F < 981:57.Suppose that (1) is true. Then we can profit by buying stock baskets using the replicat-ing strategy and shorting futures. The replication strategy will lock in the price ofacquiring each basket on the expiry date at a price of $981.57 (the loan repaymentvalue divided by the number of baskets). Meanwhile the futures price will lock in ahigher price of selling each basket resulting in sure profit. Everyone would try to pursuethis trade, forcing the futures price down or the spot price up. The opposite strategy willbe taken in case (2). We would go long futures, short the baskets at 978.80 spot, andlend money at a rate of 1.13%. We would lock in a forward sale price of 981.57 perbasket through the replication strategy and a lower price of acquiring them in thefutures market, resulting in sure profit.

Fair value with dividends

The fair value equation must be amended for stock index futures to reflect the fact thatsome of the stocks in the index may pay dividends. In the replicating strategy, thismeans that the guaranteed cost of acquiring the baskets on the expiry date, by buyingthem spot and holding them till futures expiry, will be lower by the amount of dividendsreceived between now and expiry. In other words, the cost of financing the purchasenow will be equal to the interest charge less any dividends received:

F ¼ S þ ðFinancingÞ � ðDividendsÞ


There are two ways of including dividends into the fair value or cost-of-carry equation,depending on our confidence in the future dividend estimates. For shorter expiries, wecan typically predict very accurately the dollar amounts and the dates of the dividendsD. In this case, we can present-value them to today, subtract them from the spot price,and future-value the net amount that must be borrowed to purchase the stock basket.This logic can be represented as:

F ¼ ½S � PVðDÞ�ert

For longer expiries, we may feel more confident, assuming that the dividend rate will bea constant or at least a predictable percentage of the stock price. Assuming we expressthat rate as a continuously compounded and annualized rate d, we can subtract it fromthe interest charge like this:

F ¼ Seðr�dÞt

Suppose that, in the above replicating strategy example, over the next 3 months weexpect to receive dividends whose present value we estimate to be $4 per basket. Thenwe can borrow 250� $4 less, or:

978:80� 250� 4� 250 ¼ $243,700

and use the $1,000 in present value from expected dividends in addition to the borrowedfunds of $243,700 to acquire 250 baskets at $978.80 per basket for a total cost of$244,700. The future payoff will still be the same. But, now, the futures break-evenprice that will make us indifferent between synthetic replication and futures will belower to reflect the lower cost of replication by acquiring the securities spot:

F ¼ ð978:80� 4Þe0:0113�ð1=4Þ ¼ 977:56

Instead, suppose that, in the above replicating strategy example, over the next 3months, we expect a continuously compounded, annualized dividend yield of1.6371% per basket. Then we can also borrow less than in the no-dividend case. Thefutures break-even price or fair value becomes:

F ¼ 978:80eð0:0113�0:016371Þ�ð1=4Þ ¼ 977:56

Notice in our examples that the $4 present value was chosen to be equivalent to the1.6371% continuous rate. The choice of the form of the fair value equation depends onour confidence in which is a more reliable estimate of the dividend payout: the PVamount or the dividend yield.

Single stock futures

In the early 2000s, following the lifting of a ban on such products by U.S. regulators,futures contracts on narrow indexes (custom baskets and exchange-traded funds, orETFs) and single stocks began trading on the OneChicago (ONE) and NASDAQLIFFE (NQLX) futures exchanges.2 As of late 2003, there were close to 100 singlestock futures contracts traded on each of the exchanges and fewer than 100 narrow


2 Single stock futures traded on LIFFE in London before they were introduced in the U.S.

index futures altogether. Each futures contract represents 100 shares of a stock (orETF) delivered on a stated expiry date. There are only two or at most three sequentialexpiry dates available, but the market is growing. Single stock futures compete directlywith a lucrative OTC forward market in which dealers quote forward bids and asks onindividual stocks to institutional investors.

The Wall Street Journal of July 22, 2003 quoted the following futures prices foreBay’s stock (ticker: EBAY) on the ONE exchange as of the close of the previousday (EBAY closed at 111.06):

OPEN HIGH LOW SETTLE CHG VOL OPEN INTAug 109.43 111.00 109.43 111.11 �0.96 302 800Sept 109.91 111.00 109.91 111.20 �0.87 138 1,243

Note that the increasing prices for farther maturities reflect a positive cost of carry forEBAY (no dividend and positive interest rate). Volume and open interest are still quitelow compared with major contracts, like the S&P 500, and they tend to be even lowerfor some lesser known custom indexes.

6.4 CURRENCY FORWARDS AND FUTURES

As on stock indexes and commodities, there are futures contracts on currencies tradedon major futures exchanges in the U.S., Europe, and Asia. The unique feature incurrency trading is that the market is dominated by OTC forwards, rather than stan-dardized futures, and that a significant share of the volume of transactions (about 30%as opposed to less than 5% in other markets, the rest being interbank or dealer-to-dealer activity) is linked directly to retail demand of non-financial corporations, manag-ing their foreign exchange (FX) exposure. Outright forwards represent only 7% of thetotal volume, but forwards are packaged with spots or forwards for other dates to formshort-term currency swaps, which are the main instruments traded in pure non-spot FXtransactions.

Spot and forward FX rates are quoted continuously in the interbank market and areobservable on many financial screens, like Bloomberg or Reuters. Normally, onlyforwards for standard maturities of 1-, 3-, 6-, and 12-months may be displayed, butforwards for customized expiry dates, potentially all the way out to 10-years, can beeasily arranged on most major currencies. Transaction sizes are large with $10 millionbeing a basic lot.

On August 7, 2003 a UBS website3 contained the following quotes for the USD/EURmarket:


3 Go to http://quotes.ubs.com/quotes/Language=E then click on Forex/Banknotes and Forwards USD.

Table 6.3 UBS FX forwards EUR/USD for August 7, 2003

Type Expiry date Points Bid Ask Time

Spot 1.1374 1.1381 19:31:00ON 08.08.2003 �0.36/�0.33 1.1374 1.1381 15:49:00TN 11.08.2003 �1.06/�1 1.1373 1.1380 18:42:00SN 12.08.2003 �0.34/�0.31 1.1374 1.1381 04:43:00SW 18.08.2003 �2.3/�2.21 1.1372 1.1379 19:31:002W 25.08.2003 �4.63/�4.48 1.1369 1.1377 19:03:001M 11.09.2003 �12.5/�7.5 1.1361 1.1374 13:41:002M 14.10.2003 �23.2/�18.2 1.1351 1.1363 15:08:003M 12.11.2003 �32.3/�27.3 1.1342 1.1354 18:19:004M 11.12.2003 �41.4/�36.4 1.1333 1.1345 19:31:005M 12.01.2004 �50.5/�45.5 1.1323 1.1336 19:12:006M 11.02.2004 �58.9/�53.9 1.1315 1.1327 19:29:009M 11.05.2004 �82.27/�77.27 1.1292 1.1304 19:32:001Y 11.08.2004 �103.22/�98.22 1.1271 1.1283 19:32:002Y 11.08.2005 �151/�136 1.1223 1.1245 19:30:003Y �130/�96 1.1244 1.1285 19:27:004Y �53/�7 1.1321 1.1374 19:00:005Y 53/111 1.1427 1.1492 19:12:00

http://quotes.ubs.com/quotes/Language=E then click on Forex/Banknotes and Forwards USD.Copyright UBS 1998–2003. All rights reserved. ON ¼ Overnight, TN ¼ Tomorrow/next, SN ¼ Spot/next, SW ¼ Spot/week.

Valid quotes go out 2 years (3-, 4-, and 5-year quotes were not updated that day) and allforwards are related to the spot quotes of USD/EUR 1.1374/81 (bid/ask) throughforward points. These are decimals to be added to the significant digits of the spotrate. For USD/EUR, there are four significant digits after the decimal. For example,to arrive at the 9-month forward bid, we have to take the forward points of �82:27,which are shorthand for �0.008 227, and add that to the spot bid of 1.1374 to get1.1292. Dealers shout only the forward points to each other over the phone. Just likespot, the USD/EUR forwards are quoted here in American terms (i.e., with USD beingthe pricing currency of the ‘‘commodity’’ EUR). In this example, the EUR can be saidto be trading at a forward discount as forward prices of EUR in dollars are lower thanspot prices (all points are negative). Alternatively, the USD is trading at a forwardpremium. What is special for currencies is that assets underlying the forward contractsare currencies themselves. Quotes and points measuring the premium/discount magni-tudes can thus be inverted easily to suit the viewpoint of the customer. This dual natureof FX rates also has implications for the cost-of-carry arbitrage link between spot andforward FX.

Fair value of currency forwards

The fair value equation for currency forwards is analogous to that for stocks and stockindexes. It reflects the cash-and-carry argument. Currency forwards are redundantcontracts; they can be synthesized from spot FX and interest rate transactions.Instead of locking in the FX rate at which we can exchange USD into EUR 3months from today through a forward contract, we can borrow funds in USD today


and buy the target currency, EUR, in the spot market. But just like stocks that generatecash yields in the form of dividends, the target currency, EUR, can generate a cash yieldin the form of an interest accrual on a 3-month investment in a risk-free asset denomi-nated in EUR. With stocks the dividend accrues automatically over the carry periodonce they are purchased. With currencies, we invest idle cash denominated in the targetcurrency in a short-term deposit with a maturity equal to the forward expiry date. In 3months we have an inflow of EUR from the deposit and an outflow of USD for therepayment of the USD borrowing. On the spot date, the cash flows in both currenciesnet to zero. Dollars are borrowed and used completely to purchase euros; euros arebought, but immediately deposited. The three simultaneous spot transactions (borrowUSD, exchange USD into EUR, and lend EUR) replicate exactly the redundantforward. The cost-of-carry link between forward and spot FX rates, both expressedin terms of units of currency FX1 per unit of currency FX2, can in general be written asfollows:

F

�FX1

FX2

�¼ S

�FX1

FX2

�þ ðFinancing in FX1Þ � ðFinancing in FX2Þ

Note that the last term in this equation is equivalent to the dividend term in the stockindex futures fair value. We show the discrete interest rate version of that equation inthe next section. For interest compounded continuously in both currencies, the cost-of-carry link between forward and spot FX rates reduces to:

F

�Currency1

Currency2

�¼ S

�Currency1

Currency2

�eðrCurrency1�rCurrency2 Þt

The cash-and-carry arbitrage spot–forward link for currencies is referred to by financialeconomists as covered interest rate parity.

Covered interest-rate parity

Suppose we observe that today’s JPY/EUR spot FX rate stands at 100. At the sametime, 1-year deposit rates are 2% in yen and 4% in euros. The covered interest rateparity (CIRP) principle states that the 1-year forward rate must be equal to:

F

�JPY

EUR

�¼ S

�JPY

EUR

�� 1þ rJPY

1þ rEUR¼ 100

1:02

1:04¼ 98:0769

This is guaranteed as long as borrowing and lending in each currency is unrestrictedand there are no FX controls. The argument is replication or arbitrage. In the replica-tion strategy, we can synthesize a long forward by borrowing in yen, selling yen spot foreuros, and investing idle euros till the expiry. We can synthesize a short forward byborrowing in euros, selling euros spot for yen, and investing idle yen till the expiry. Ifboth sides of the forward can do it on their own synthetically, then the forward is aredundant contract. The forward rate agreed on by both parties must reflect the cost ofthe replication strategy.

The arbitrage argument underscores the fact that the forward FX rate of 98.0769 isnot just a mathematical fiction, but will be ensured by dealers seeking riskless profit.Suppose a dealer quotes a forward of 98.0000. This is lower than the theoretical CIRPrate. The interpretation can be that the EUR can be bought for less in the forward


market than through a spot-and-carry transaction. The arbitrage strategy will thusinvolve buying euros forward and selling them spot. In order to offset FX cashflows, we will have to borrow euros today, agree to pay interest on the borrowing,and deliver the borrowed euros to the spot FX transaction. We will deposit theobtained yen, earn interest on the deposit, and deliver the yen from the maturingdeposit into the forward FX contract to sell yen for euros. Let us say that an‘‘errant’’ dealer quoted the forward of 98.0000 for the maximum size of EUR10,000,000. Here is how we profit:

. Borrow EUR 9,615,384.62 at a rate of 4% for 1 year.

. Spot sell EUR 9,607,843.14 at JPY/EUR 100 for JPY 960,784,313.73.

. Deposit JPY 960,784,313.73 at a rate of 2% for 1 year.

. Enter into a forward contract to sell JPY 980,000,000 at a rate of JPY/EUR 98 forEUR 10,000,000.

Net cash flows in yen are zero as we purchase yen in the spot FX market and immedi-ately lend out the entire purchase. In euros we borrowed more than we sold in the spotFX market, resulting in a positive cash flow of EUR 7,541.48. In 1 year, we have netzero cash flows in both currencies:

. We collect the yen deposit, which accrued to: 960,784,313.73� 1.02 ¼ JPY980,000,000.

. Deliver JPY 980,000,000 to the forward contract to receive EUR 10,000,000.

. Pay off the borrowing, which accrued to: 9,615,384.62� 1.04 ¼ 10,000,000.

We keep the EUR 7,541.48 as riskless profit. Note that by going long the forward topurchase euros we are bidding the price of euros in yen up, and by selling euros spot weare putting a downward pressure on spot euro and bidding up spot yen. We are alsoputting an upward pressure on EUR deposit rates and downward pressure on JPYdeposit rates. All these four effects would tend to bring the prices into a parity withrespect to each other as defined by the CIRP. The most obvious effect is that the errantdealer may adjust his forward quote up after seeing our demand for forward euros. Thereader should be able to determine how to profit, had the dealer erred in the oppositedirection and shown a forward rate higher than that posited by the CIRP.The CIRP equation for intra-year periods needs to be amended to reflect the account-

ing for interest using money market rates. For example, for 3-month strategies usingLIBOR rates the JPY/EUR example would look like this:

F

�JPY

EUR

�¼ S

�JPY

EUR

��

1þ LJPYAct

360

1þ LEURAct

360

Let us apply the logic of that equation to the FX rates for August 7, 2003 as quoted byUBS in Table 6.3. The spot ask FX rate is USD/EUR 1.1374 bid and USD/EUR 1.1381ask. Within the same UBS site, we could find the 3-month LIBOR rates: 1.14 for USDand 2.1613 for EUR. We will ignore the fact that LIBOR quotes and FX quotes maynot be from the same exact time of the day. Using 97 as the actual number of days


between August 7 and November 12, the stated expiry of the 3-month FX forward, weobtain the fair forward bid rate:

F

�USD

EUR

�¼ 1:1374�

1þ 0:011497

360

1þ 0:021 61397

360

¼ 1:1343

and for the fair ask rate:

F

�USD

EUR

�¼ 1:1381�

1þ 0:011497

360

1þ 0:021 61397

360

¼ 1:1350

The actual forward bid–ask quotes of USD/EUR 1.1342/1.1354 contain a wider spread,reflecting the potential mark-up charged by UBS for unsolicited orders. They alsoensure that on August 7, 2003 we could not have ‘‘picked UBS off’’ to earn risklessprofit. The interbank market has already arbitraged out all riskless profit opportunitiesensuring that the cash-and-carry CIRP holds!

Currency futures

Currency futures trade on two U.S. exchanges. On the CME, all currency pairs areagainst the U.S. dollar with foreign currency as the underlying commodity and thedollar as the pricing currency; on the FINEX, there are a few cross-pairs, all against theeuro, with the euro as the underlying commodity and the other currency as the pricingcurrency of denomination. Contract sizes are large by retail standards, but relativelysmall by wholesale market standards. For example, on July 22, 2003 the Wall StreetJournal reported the following settlement prices for the previous business day:

LIFETIME OPENOPEN HIGH LOW SETTLE CHG HIGH LOW INT

Japanese yen (CME)—Y¼12,500,000 ($ per Y¼)Sept .8437 .8471 .8432 .8448 �0.0010 .8815 .8220 81,966Dec .8478 .8486 .8465 .8471 �0.0010 .8915 .8350 20,435Est vol 8,659; vol Fri 19,005; open int 102,436, �4,422

. . .Euro/US dollar (CME)—c¼ 125,000 ($ per c¼ )Sept 1.1239 1.1338 1.1226 1.1326 .0057 1.1896 .8780 92,581Dec 1.1265 1.1310 1.1213 1.1299 .0057 1.1860 .9551 1,550Mr04 1.1218 1.1270 1.1218 1.1275 .0057 1.1795 1.0425 253Est vol 33,046; vol Fri 41,304; open int 94,456, �2,557

On that same day, the spot FX rate for USD/EUR was 1.1347, and the U.S. dollar andeuro LIBOR rates stood at:

Dollar Euro1-month 1.1000% 2.1205%3-month 1.1100% 2.12275%6-month 1.1200% 2.10938%12-month 1.21125% 2.11313%


We can use the CIRP equation:

F

�USD

EUR

�¼ S

�USD

EUR

��1þ LUSD

Act

360

1þ LEURAct

360

to come up with the fair value of the futures. Using interpolated interest rates for 2-, 5-,and 8-month expiries, we get for September:

F

�USD

EUR

�¼ 1:1347�

1þ 0:011 050 02

12

1þ 0:021 216 252

12

¼ 1:13278

For all three delivery months we compute:

Sept 1.13278Dec 1.13003Mr04 1.12754

Note that LIBOR rates and futures settlement prices are not from the same times(LIBOR settles from 3p.m. GMT and futures 3 p.m. CDT), so our calculation producesa small error (which is the greatest for the shortest maturity). All numbers are withinthe low and the high for the day.Let us briefly review the mechanics of daily settlement for currency futures. These are

identical to those for commodities with the currency in the denominator treated as theunderlying commodity. Suppose on July 21 we entered into five long December USD/JPY contracts right at the close at .8471. Note that for JPY, the .8471 price is in dollarsper 100 yen, or it is shorthand for .008471 in USD/JPY, which is equivalent to JPY/USD 118.05. Suppose by the next day the JPY/USD settlement rate changed to 119.25,or equivalently to USD/JPY .008386 or USD/100JPY .8386. We have a negative cashflow in the form of a variation margin settlement of:

5� 12,500,000ð0:8386� 0:8471Þ=100 ¼ $� 5,328:48

The multiplier for the price as stated in the contract is thus 125,000. Note that for theUSD/EUR contract, the multiplier is simply equal to the size of the contract, which is100,000 as the price is stated in dollars per 1 euro. It can be shown in a manneranalogous to the commodity arguments that going long five December JPY contractsallowed us to lock in an exchange rate of 118.05 on JPY 62,500,000 for delivery onDecember 19, 2003. The net variation margin accumulated by that expiry date willoffset any difference between the spot FX rate on that date and the original rate of118.05 on the purchase of JPY 62,500,000.

6.5 CONVENIENCE ASSETS—BACKWARDATION

AND CONTANGO

Financial assets, like stocks, gold, fixed income instruments, and currencies, are ex-amples of pure assets. They are held purely for investment purposes. Investors are


indifferent between holding the assets themselves and claims on them. For example,most stock investors hold shares in their brokerage accounts in street name. They rarelyask for stock certificates in order to exercise their voting rights. What matters to them isthat they receive dividends and that they are able to sell their positions on demand. Inthe meantime, brokerage houses are free to lend these shares to others who can thenshort-sell them. As long as there are people holding these assets for investment purposes(i.e., they are long these assets), there will be no restrictions on short sales and noadditional costs to shorting. For pure assets, both cash-and-carry and reverse cash-and-carry strategies are executable and at approximately the same cost. This is perhapsthe most obvious with currencies where a purchase of one currency automaticallymeans a sale of another. For pure assets, the general form of the fair value equationlooks like this:

F ¼ S þ ðFinancingÞ þ ðStoringÞ � ðCash yieldÞwhere Financing ¼ interest paid on funds used to purchase the asset spot or interestearned on proceeds from a short sale of the asset spot; Storing ¼ storage cost equal to 0for most financial assets except gold; and Cash yield ¼ dividends or coupon interestfrom the underlying asset (e.g., dividends from stocks and interest from currencies orbonds). We have already seen some specific guises of this equation.

Most agricultural, energy, and metal commodities are not pure, but rather con-venience assets. Holders of these commodities own them not only for investmentpurposes, but to be used in production. Any disruption in that production may becostly. There may thus be an additional value, which varies from user to user, toowning the physical asset. That value, called a convenience yield, can be most easilyviewed as an additional cash yield or dividend; the owner of the asset has a valuableinsurance policy that his business will not be disrupted. When lending out the asset tobe shorted spot by someone else, the owner has to be compensated for the loss of thisconvenience. The reverse cash-and-carry strategy may not be executable at all or maylead to very high costs. The general fair value equation for futures and forwards has tobe amended to include the convenience yield, which reflects the implicit value of theconvenience benefit to the marginal investor in the market (the most willing to lend theasset out in sizable quantity):

F ¼ S þ ðFinancingÞ þ ðStoringÞ � ðCash yieldÞ � ðConvenience yieldÞFor most commodities, the convenience yield tends to be large resulting in subsequentfutures prices to be smaller than the previous ones and the spot price. This is callednormal backwardation. There are many economic explanations for the predominance ofbackwardation in commodity futures, the most popular being that most hedgers areproducers of commodities who tend to short futures at a discount to fair value in orderto compensate speculators for accepting the risk.

There are times when the convenience value is small relative to the financing cost netof storage, resulting in a situation where futures prices increase with maturity. This iscalled a contango. Note that financial assets are more likely to be in this category aslong as the cost of financing exceeds any cash yield. There are also special situationsleading to a contango (e.g., resolutions of wars, expectations of new supply lines, etc.).We can also come up with examples of contangos due to negative convenience yields,like holding wheat right before a plentiful harvest.


6.6 COMMODITY FUTURES

Most commodities in the world are priced in U.S. dollars. The exchanges for rareagricultural or mineral products, or for local physical delivery points, can be scatteredaround the world (e.g., groundnuts and safflower oil in Mumbai, or olein in Jakarta),but by far the main centers for futures trading are Chicago, New York, and London.On the exchanges, producers and users of commodities can hedge their price risks inagricultural, petroleum, and metal commodities. While only a small percentage oftrading activity (5%) is related to hedging and the rest is related to speculation, theexistence of speculators is fundamental for price risk sharing. In each commoditymarket, there tends to be an imbalance of economic power between producers andusers which is filled by the speculators. The most typical situation in a commoditymarket is normal backwardation whereby futures prices decrease with the contractmaturity date due to a convenience yield of owning the physical asset. On July 22,2003 the Wall Street Journal reported the following information for the crude oilfutures traded on the NYMEX exchange:


Crude oil, light sweet (NYM)—1,000 bbl ($ per bbl)Aug 32.00 32.10 31.25 31.78 �0.18 32.35 21.16 45,102Sept 31.05 31.17 30.33 30.83 �0.20 31.53 21.05 190,648Oct 30.52 30.64 29.95 30.40 �0.09 30.93 20.55 55,331Nov 30.03 30.05 29.50 29.93 �0.01 30.40 20.70 28,262. . .Dc07 23.80 23.80 23.80 24.04 þ0.18 24.20 19.50 2,590Est vol 191,187; vol Fri 156,891; open int 536,775, þ11,528.

On that date, West Texas Intermediate Oil traded spot at 31.98. Let us use the estimatefor financing cost based on LIBOR as in the stock index futures example (i.e., 1.13%continuously compounded) and assume that storage costs run $2 per barrel per year.This is equivalent to 6.10% continuously compounded. We can compute the fair valueof futures without the convenience yield using a continuous formula:

Ffair ¼ Seðr�syÞt

where sy ¼ storage cost expressed as continuous yield give-up. We can also back outthe convenience yield, by solving for cy such that:

Factual ¼ Seðr�sy�cyÞt

Applying the formula for 1-, 2-, 3-, and 4-month delivery periods we get the followingresults:

Table 6.4 Fair values and convenience yields forNYMEX oil futures as of July 22, 2003

Month Fair cy

Aug 31.85Sep 31.72 17.00Oct 31.59 15.30Nov 31.45 14.90


The results show a pretty substantial convenience yield value for the near months. Asthis number is difficult to determine from fundamentals and dominates the easilyestimatable financing and storage costs, it renders the spot–futures and futures–futures cash-and-carry arbitrage highly speculative. The convenience yield drops withmaturity to less than 3.5% for the December 2007 contract (using r ¼ 3:13), reflectingthe fact that futures–futures arbitrage involving farther months is mostly financial(borrowing for Dec 2007 and lending for Dec 2006 or vice versa), with future conve-nience value severely discounted at this point.

On July 22, 2003, the news of the day was a substantial drop in coffee prices asexpectations of frost affecting the August crop in Brazil faded. Traders dumped coffeein the spot markets. This resulted in a contango for coffee prices which were reported tobe:


Coffee (CSCE)—37,500 lb (cents per lb)July 60.00 60.00 60.00 59.15 �3.15 79.00 55.80 201Sept 62.85 63.00 59.70 60.25 �2.95 80.50 57.30 44,656Dec 65.25 65.40 62.00 62.85 �2.75 81.50 59.50 13,222. . .Dc04 73.50 73.50 73.00 72.55 �2.25 80.50 71.50 1,490Est vol 16,058; vol Fri 6,298; open int 69, 251, �343.

Assuming zero storage costs, the September–July price difference reflected the con-venience yield of minus 9.92%.

6.7 SPOT–FORWARD ARBITRAGE IN INTEREST RATES

In this section, we show how futures and forwards can be used in lieu of spot zero-coupon instruments in zero-vs.-coupon strategies of coupon stripping and replication.This leads us directly to establish tight arbitrage relationships between interest ratefutures/forwards and coupon and zero-coupon spot rates.

Recall from Chapter 2 that purchases and sales of zeros can be synthesized frompurchases and sales of coupon instruments, and vice versa. For example, the purchaseof a 3-year semi-annual coupon bond can be replicated by the purchase of six zero-coupon bonds, with face values equal to the coupon amounts and maturities matchedto the coupon dates, and one zero-coupon bond matched with the principal repayment.We argued that the price of such a package of zero-coupon securities must be identicalto that of the coupon instrument. Similarly, we argued that any zero-coupon instru-ment can be replicated by the right combination of coupon instruments and other zeros.For the remainder of this section we will restrict our focus to the relationship of zeros toforwards and futures, and skip the obvious extension of the arbitrage equations ofcoupon instruments as packages of zeros and, thus, by extension as functions offorwards and futures. We also defer the discussion of credit spread issues by assumingthat zero-coupon bonds, spot and futures LIBOR deposits, forwards, and futures areall entered into between parties of the same credit quality.


Synthetic LIBOR forwards

We start with a rather concocted example that will lead us later to more advancedtopics. Suppose a customer appears at the ABC Bank to ensure that the bank will lendhim $100 million 1 year from today for 1 year. The customer is willing to let the rate onthe loan float for 1 year. He will be charged the spot rate 1 year from today and willrepay the principal and interest based on that rate at the end of the interest-accrualperiod 2 years from today. The bank, in order to ensure the availability of funds asprovided for by the contract with the customer, may choose to engage in the followingstrategy with third parties (assume a simple day-count):

. Borrow for 2 years at today’s 2-year zero-coupon rate of 6.5163%.

. Lend for 1 year at today’s 1-year zero-coupon rate of 6.0000%.

Let us compute the face values of the borrowing and lending by working backwards:

. In order to have $100 million on hand 1 year from today, ABC will have to lendtoday:

100,000,000� ½1=ð1þ 0:06Þ� ¼ $94,339,622:64

. ABC will have to borrow that amount for 2 years (i.e., it will have to agree to repayin 2 years):

94,339,622:64� ð1þ 0:065163Þ2 ¼ $107,035,114:80

By borrowing $94,339,622.64 today for 2 years at 6.5163% to repay $107,035,114.80 in2 years and lending $94,339,622.64 today for 1 year at 6.0000% to receive $100,000,000in 1 year, the bank locks in a certain cost of borrowing funds. That cost is equal to therate r such that:

100,000,000� ð1þ rÞ ¼ 107,035,114:80

We solve to get r ¼ 7:0351%. In 1 year’s time, when the customer comes to collect thepromised $100 million, ABC agrees to charge him the then-prevailing LIBOR. So, forthe accrual period, the bank will pay a rate of 7.0351 to a third party and receiveLIBOR from the customer. This set of cash flows is equivalent to entering a 12� 24FRA as a payer of fixed on $100 million. If the FRA market did not exist, then LIBORforwards could with some effort be synthesized by borrowing and lending in the zeromarket for the right terms.In our example, the bank had two other choices:

. Do nothing; wait till 1 year from today, borrow at LIBOR, and provide the funds tothe customer at LIBOR at a complete wash (i.e., net zero cash flows today, 1 yearfrom today, and 2 years from today).

. Lock in the lending rate to the customer, by entering a 12� 24 FRA to receive fixedon $100 million against 12-month LIBOR; the LIBOR received from the customerwould then offset that paid into the FRA and the bank would net receive fixed on theloan-cum-FRA bundle.

In the latter case, the rate fixed in the FRA would have to be 7.0351% in order for thebank to have net zero cash flows at all three points in time.We can extend the preceding example to show that any forward lending (receiving on

a FRA) can by synthesized in the zero-coupon market by borrowing for the short term


and lending for the long. That strategy is equivalent to an outflow (repayment of theborrowing) on the forward start date and an inflow (receipt of the deposit plus interest)on the forward end date (i.e., a loan (out)). The mathematics of the synthetic FRA forsimple annual periods reflect the spot borrowing and lending:

ð1þ zn1Þn1ð1þ fn1�n2Þn2�n1 ¼ ð1þ zn2Þn2where zn is a spot borrowing/lending rate and fn1�n2

is an n1 � n2 FRA rate. In theabove example, we had n1 ¼ 1 year, n2 ¼ 2 years. The equation looked like this:

ð1þ 0:060 000Þð1þ f1�2Þ ¼ ð1þ 0:065 163Þ2

Solving, we get f1�2 ¼ 0:070 351. To determine a forward rate for the period of 2 yearsstarting in 3 years knowing the 3-year zero and the 5-year zero, we would have to solvefor f3�5:

ð1þ z3Þ3ð1þ f3�5Þ2 ¼ ð1þ z5Þ5

The equations are more complicated for intra-year periods and when the right day-counts are used, but in a predictable way. For example, suppose that we know a 6-month zero compounded quarterly z6 and a 9-month zero compounded quarterly z9,both on an Act/360 basis. Then the 6� 9 forward could be derived from:�1þ z6 � Act0�3

360

��1þ z6 � Act3�6

360

��1þ f6�9 � Act6�9

360

�

¼�1þ z9 � Act0�3

360

��1þ z9 � Act3�6

360

��1þ z9 � Act6�9

360

�

where we substitute the correct number of days in each interest compounding period.

Synthetic zeros

We can reverse the above relationships to synthesize zeros from forwards. If a 3� 6forward pay is a combination of a long 3-month zero (lending) and a short 6-monthzero (borrowing), then we can also construct a long 6-month zero (lending for the longterm), by a long 3-month zero (lending for the short term) and receiving on a forward(guaranteeing the lending rate for the intervening 3� 6 period).

Suppose a customer of the ABC Bank requests a $100 million spot loan for a 2-yearperiod. ABC charges 6.00% on 1-year loans (equal to its own cost of borrowing fundsfor 1 year). It also realizes that 12� 24 FRAs are quoted at 7.0351%. What rate shouldit charge the customer?

In order to write its customer a check for $100 million today in exchange for a 2-yearIOU, ABC can borrow funds for 1 year at 6.00% and can lock in the rate on a rolloverof that borrowing at 7.0351% by paying fixed on a FRA or shorting ED futures. Theborrowed $100 million will have to be repaid in 1 year with:

100,000,000� ð1þ 0:06Þ ¼ $106,000,000

ABC can agree to pay 7.0351% on a 12� 24 FRA with a notional principal of $106million or short 106 12-month ED futures contracts (the futures strategy translates inreality to a string of 3-month LIBOR futures with expiries in 12, 15, 18, and 21months). In 1 year’s time, in order to repay the original $100 1-year borrowing, ABCwill have to borrow $106 million for one year at the then-prevailing 12-month LIBOR


rate. But, we know from our previous discussion that no matter what the 12-monthLIBOR is 1 year from today, ABC will have been guaranteed a net interest charge at therate of 7.0351%. Thus its borrowing will accrue over the second year to:

106,000,000ð1þ 0:070 351Þ ¼ $113,457,206

In order for the ABC Bank to break even, it has to charge its customer on a 2-year spotloan of $100 million a rate z2 such that:

100,000,000� ð1þ z2Þ2 ¼ 113,457,206

Solving, we get z2 ¼ 6:5163%.The above relationships hold for zeros and forwards of all maturities, whether short

or long. We can use that fact to construct synthetic zeros in a variety of different ways.If a 2-year zero can be constructed using a 1-year zero and a 1-year-by-2-year forward,then a 3-year zero can be constructed either as:

. A 2-year zero and a 2-year-by-3-year forward.

. Or a 1-year zero, a 1-year-by-2-year forward and a 2-year-by-3-year forward.

Let us continue with the numerical example. Suppose that prior to signing the 2-yearloan agreement, our customer changes his mind and asks for a 3-year $100 million loaninstead. ABC can turn to the 24� 36 FRA market, currently trading at 7.4674%, tolock in the rate for the forward 1-year period starting in 2 years on a $113,457,206principal. It will have to borrow that amount in the spot market, but its net rollover ratewill be guaranteed. In order to satisfy its borrowing repayment 3 years from today, itwill have to receive from its customer exactly:

113,457,206� ð1þ 0:074 674Þ ¼ $121,929,528:70

That means, it will quote today a 3-year zero-coupon rate z3 such that:

100,000,000� ð1þ z3Þ3 ¼ 121,929,528:70

or z3 ¼ 6:8324%. The rollover strategy for the first 2 years is at the bank’s discretion.ABC can either:

. Borrow $100 million in the 2-year zero market at 6.5163%.

. Or borrow $100 million in the 1-year zero market at 6.00% and roll it over in the1-year-by-2-year forward market at 7.0351%.

The result will be the same: $113,457,206 owed after 2 years. This will be rolled over atthe guaranteed (upfront) rate of 7.4674% for the third year.It should be clear by now that any zero can be synthesized by a zero of shorter

maturity and a string of forwards for the intervening period.

Floating-rate bonds

A floating-rate bond is a bond whose coupon is not known in advance, but is tied to thefuture performance of an index or a rate. In the standard floating-rate bond, thecoupon accrual period is matched to the maturity of the short-term interest rate usedto determine the coupon. The short-term rate is set at the beginning of each couponperiod and the coupon based on that rate is paid at the end of the coupon period. The


coupon payment is computed as the set rate times the face value of the bond times theappropriate day-count fraction. Compared with a fixed rate bond, the valuation of thefloating rate bond may appear difficult as we do not know the cash flows of the bonds inadvance. However, the forward certainty equivalence principle makes valuation of mostfloating rate bonds straightforward.

Let us consider a 10-year, $100 face value, annual coupon bond whose floating rate istied to the 12-month LIBOR. The coupon rate is known only for the first year as it isbased on today’s 12-month deposit rate. The rate for the second year, not known today,will be based on the LIBOR rate 1 year from today and paid 2 years from today. Therate for the third year, not known today, will be based on the LIBOR rate 2 years fromtoday and paid 3 years from today. And so on. There are two ways to look at this bondtoday, exactly 1 year prior to the next cash flow.

One is to recognize that a floating rate bond is equivalent to an artificial rolloverstrategy. The issuer borrows $100 for 1 year. The fair rate on that borrowing is today’sLIBOR which is set to be today’s coupon for the first year. In 1 year’s time, the issuerborrows $100 again for 1 year in order to repay the original loan of $100 obtained attime zero for 1 year. He also pays the coupon interest equal to time zero LIBOR. Therate set for year 2 is equal to the 1-year LIBOR 1 year from today which is a fair rate fora new 1-year loan. The issuer then proceeds with rolling over the loan every year, eachtime at a new, fair, 1-year rate. Each year, right after the loan is rolled over, the value ofthe issuer’s IOU for the next year is equal to $100. He promises to repay the principal of$100 and the interest on that principal set at that time; this is to be paid a year later. Thetotal discounted back by 1 year at the prevailing 1-year rate is $100.

Another way to look at the floating rate bond is to recognize that both the issuer andthe buyer of the bond have the option to lock in, at no cost, the uncertain cash flows ofthe bond into their certainty equivalents through the FRA market or the ED futuresmarket. A buyer of the bond could convert the unknown future LIBOR payments intofixed cash flows, which would most likely be different each year, but known in advanceand equal to today’s FRA rates spanning the respective coupon periods. Using thenumbers from our previous example, the investor can lock in the following couponpayments for the first 3 years:

. For year 1, today’s LIBOR or zero rate of 6.00%.

. For year 2, the 1-year-by-2-year forward rate of 7.0351%.

. For year 3, the 2-year-by-3-year forward rate of 7.4674%.

And so on. A floating rate bond is thus equivalent to a bond for which the coupon ratesare fixed today, but vary each period. If the floating rate bond has a spread (i.e., isquoted as, say, LIBOR plus 1.5%), then such floating rate bond is equivalent to a sumof two fixed coupon bonds: one with varying coupons equal to the forward rates andone with all coupons equal to the spread of 1.5%, net of the short zero to offset thedouble-counted principal.

Synthetic equivalence guaranteed by arbitrage

The equivalence of spot zero strategies with synthetic forwards leads to the followingsummary of important findings:


. The payoff on a Eurocurrency forward (FRA) contract can be synthetically repli-cated by borrowing and lending the appropriate amounts in the zero-coupon market.

. Since forwards and futures are almost perfect substitutes, the payoff on a Euro-currency futures contract can be synthetically replicated by borrowing and lendingthe appropriate amounts in the zero-coupon market.

. Given the forward and futures certainty equivalence and that the market places nocost on converting future spot to today’s futures or forwards, a bank will not chargethe customer for locking in the future unknown loan rate.

. Longer term zero rates can be costlessly synthesized from shorter-term zeros andintervening forwards.

. Longer term zero rates can be costlessly synthesized from shorter-term zeros andintervening futures.

. Coupons, zeros, floating-rate bonds, forwards and futures are all synthetic equiva-lents of the appropriate combinations of longs and shorts with each other (i.e., allcan be replicated at no cost with packages of the other instruments).

. The possibility of arbitrage profits ensures that the mathematical relationshipsimplied in the above statements hold.

6.8 CONSTRUCTING THE ZERO CURVE FROM FORWARDS

In Chapter 2, we showed how quotes for coupon bonds can be used sequentially tocompute discount rates for present value calculations through a process called a zerocurve bootstrap. In this section, we revisit the construction of the zero curve, this timewith the use of futures and forwards. Like before, we will use observed quotes inchronological order in order to create a set of rates or, alternatively, discountfactors, appropriate for computing the present values of cash flows set for futuredates. This is a fundamental task in any valuation process of not only money marketinstruments, bonds, structured securities and swap derivatives, but also in equity,currency and commodity forwards and options.The great insight of Chapter 2 was that coupon securities are packages of zeros. Let

us assume semi-annual coupons. By knowing the yield on a 6-month coupon weautomatically know the rate for a 6-month zero. A 12-month coupon was a packageof two zeros: one with 6-month maturity and a face value equal to the 6-month couponcash flow, and one with 12-month maturity and a face value equal to the 12-monthcoupon cash flow plus the principal. If we knew the ‘‘blended’’ yield on the couponsecurity and the yield on the first zero in the package, then we could back out the yieldon the second zero, since the price of the package had to equal the sum of the parts.Similarly by knowing the price of and the yield on a 18-month coupon security andpreviously computed yields of the 6- and 12-month zeros, we could back out the yieldon the 18-month zero. We could continue like that all the way to the coupon bond withthe longest maturity, creating a set of zero-coupon yields. Using zero yields, we couldthen discount cash flows scheduled for any future dates and compute the prices of anyfixed income security.The coupon bootstrap is typically used in constructing the curve for dates past the

1-year or 5-year point to the 30-year point. It is considered too crude for shortermaturities, as our first observation point is 6 months out for U.S. semi-annualcoupon bonds and often 12 months out for European and Canadian markets where


bonds typically pay coupons annually. A forwards- or futures-based bootstrap forshorter dates dovetails nicely with the coupon bootstrap further out. It providesliquid futures points every 3 months for 10 years for U.S. markets (CME EDfutures) and liquid futures points every 3 months out to the 3-year or 5-year markfor the pound and the euro. Finer observations can be obtained from 1-month LIBORfutures and FRAs whose dates do not correspond to the futures calendar. The choice ofwhere the two methods meet, the short bootstrap based on forwards, and the longbootstrap based on coupons depends on the relative liquidity of the instrumentsused. Their liquidity ensures that packages can be easily constructed to create syntheticsecurities. That is, it ensures that the whole process is not just a mathematical exercise,but a market reality guaranteed by potential arbitrage. Let us turn to the details of theforwards/futures-based bootstrap of the zero curve.

We are going to proceed in a manner similar to the coupon case. We will use the firstzero rate known and then forwards and/or futures with ever-increasing start and enddates to construct a discount curve of zero-coupon yields or discount factors. As it isgood practice to avoid confusion with day-counts, we will compute discount factorsand not yields for different dates. Discount factors, which are present value equivalentsof $1 received on a given future date, are unambiguous. Any rate with an arbitrarymaturity, compounding, and day-count can be recovered from them.

We use the futures quotes from the October 25, 2001Wall Street Journal. In addition,we use three spot zero rates as starting points for our curve. The maturities of these spotrates match the expiry of the first three futures contracts (i.e., November, December,and January). We will only use the November and January quote to compute discountfactors for November and January, and for periods extending 3 months after that (i.e.,February and April). The December spot will be used to compute the discount factorsfor December and for all periods ending in 3-month intervals following the quarterlyfutures expiries of March, June, September, and December. For simplicity, we assumethat the LIBORs implied in the futures contracts are for forward deposits spanningroughly 3 months from the futures expiry to the next contract’s expiry, and not for90-day deposits as assumed by the $25 multiplier. The quote information is summarizedin Table 6.5.

Table 6.5 Spot LIBORs and futures prices for October 25, 2001. Discount factors computed forforward start dates and for spot date to forward end date

Start date End date Days in Actual Yield Forward Spot discountforward period quote (Act/360) discount factor factor

Spot15-Oct-01 21-Nov-01 37 2.05 2.05 0.997 897 0.997 89715-Oct-01 19-Dec-01 65 2.12 2.12 0.996 187 0.996 18715-Oct-01 16-Jan-02 93 2.15 2.15 0.994 477 0.994 477

Futures21-Nov-01 19-Feb-02 90 97.81 2.19 0.994 555 0.992 46419-Dec-01 20-Mar-02 91 97.85 2.15 0.994 595 0.990 80216-Jan-02 19-Apr-02 93 97.87 2.13 0.994 528 0.989 03420-Mar-02 19-Jun-02 91 97.77 2.23 0.994 395 0.985 24819-Jun-02 18-Sep-02 91 97.48 2.52 0.993 670 0.979 01218-Sep-02 18-Dec-02 91 97.09 2.91 0.992 698 0.971 86318-Dec-02 19-Mar-03 91 96.60 3.40 0.991 479 0.963 582


We are given three zero yields: 2.05, 2.12, and 2.15. We are also given futures quotes.These allow locking in the rates for the periods starting on the futures expiry date andending 3 months later. We compute the actual number of days in each forward periodspanned by the futures quote and the yield implied for each of those periods as 100minus the futures price. We also compute forward discount factors (the first threeusing the spot zero rates are actually spot factors with the start date equal to the spotdate of today). These are present values as of the start date of $1 received on the enddate using the futures-implied yield. The logic follows from arbitrage as we would beable to synthesize a $1 face value loan for the forward period by locking in the rate inthe futures, and borrowing or lending at future LIBOR. From forward discountfactors we can compute spot discount factors that are present values as of today,October 25, 2001, of $1 received on the end date. These are products of appropriateforward discount factors spanning successive periods. For example, the spot discountfactor of 0.979 012 for September 18, 2002 is the product of the spot factor of0.996 187 for December, the forward factor of 0.994 595 for December–March, theforward factor of 0.994 395 for March–June and the forward factor of 0.993 670 forJune–September. From the discount factors we are able to recover discount ratesbased on any convention. As is traditional, we show the continuously compoundedyields (also unambiguous) which have the nice property that the spot continuous yieldis an average of the appropriate forwards. We also show simple interest Act/360yields.

Table 6.6 Continuous and simple yields implied by the computed spot discount factors

End date Days from spot Spot discount factor Continuous yield Simple yield

(Act/360)

21-Nov-01 37 0.997 897 2.0763 2.0500

19-Dec-01 65 0.996 187 2.1453 2.1200

16-Jan-02 93 0.994 477 2.1738 2.1500

19-Feb-02 127 0.992 464 2.1741 2.1525

20-Mar-02 156 0.990 802 2.1620 2.1423

19-Apr-02 186 0.989 034 2.1637 2.1459

19-Jun-02 247 0.985 248 2.1961 2.1822

18-Sep-02 338 0.979 012 2.2906 2.2833

18-Dec-02 429 0.971 863 2.4283 2.4295

19-Mar-03 520 0.963 582 2.6040 2.6166

Note that by definition the first three simple yields match the inputs.We have thus constructed the yield curve for the first 520 days. It should be clear that

we could have substituted FRAs for futures or used them in addition to futures forsome of the in-between dates. We also could have used more coupon instrument datesor know spot points. The result is the graph of the zero-coupon yield curve for October25, 2001 (Figure 6.1).


0

1

2

3

0 50 100 150 200 250 300 350 400 450 500 550

Days from spot

Co

nti

nu

ou

s y

ield

Figure 6.1 Zero-coupon curve.

Once we have the yield curve, the valuation of fixed income securities with no optionsembedded in them becomes trivial. Suppose someone offers us a structured note prom-ising to pay $3,500 twice, 182 days from today and 365 days from today, plus theprincipal of $100,000. How much would we pay for it?

Since we do not have the exact discount factors for 182 days and 365 days, we use thearithmetic interpolation of the continuous yields to obtain these two factors. For 182days, we use the rates for 156 days and 186 days to get the interpolated yield of 2.1635,which translates into a discount factor of 0.989 270. For 365 days, we use the rates for338 days and 429 days to get the interpolated yield of 2.3314, which translates into adiscount factor of 0.976 955. The present value of the note’s cash flows is then equal to:

3,500 � 0:989 270þ ð3,500þ 100,000Þ � 0:976 955 ¼ $104,577:33

We would be willing to pay 104.5773 of the face value for the note.

6.9 RECOVERING FORWARDS FROM THE YIELD CURVE

Once the yield curve is constructed (i.e., we have obtained a set of discount factors forall future dates), we can not only value securities with known future cash flows, but alsothose whose cash flows float with LIBOR rates (floating rate bonds, inverse floaters,leveraged floaters, etc.). We can do that because we can recover the forwards implied bythe yield curve and substitute them for future unknown LIBORs using the certaintyequivalence argument. Since all future spot LIBORs can be costlessly converted totoday’s forwards, the present value of an artificial security with forwards substitutedfor the unknown future LIBORs must be the same as that of the actual security withunknown cash flows.


The valuation of a floating-rate bond

Let us consider the following example. On October 25, 2001 we are offered a 1-yearfloating rate note with a face value of $200,000 whose quarterly coupon is equal to the3-month LIBOR. The coupon payment dates are January 25, April 25, July 25, andOctober 25, 2002, set 3 months prior. How much are we willing to pay for it? Theanalysis of the bond is summarized in Table 6.7.For the given set and pay dates, we compute the numbers of days from spot and the

number of days in each accrual period. We interpolate the continuous yields andcompute the interpolated present value (PV) factors to each set and pay date. Fromeach pair of PV factors, we can compute the implied LIBOR set (i.e., today’s forward 3-month LIBOR) for the set date. These are obtained from the arbitrage equation:

PVð$ on pay dateÞ ¼ PVð$ on set dateÞ � 1

1þ fset date�pay dateAct

360

Each cash flow is set, based on the implied LIBOR forward on the set date, to be equalto:

200,000� fset date�pay date � Act

360

and paid on the pay date, 3 months later. We can then compute the PV of each cashflow by multiplying each cash flow by the discount factor to the pay date. We sum thePVs to obtain the price of the note. This turns out to be exactly $200,000. This shouldbe no surprise given our previous argument about the nature of the floating rate note asa revolving loan always worth par.

Including repo rates in computing forwards

This generic arbitrage analysis is very general and applies to all credit markets. There is,however, one situation where the analysis has to be modified to take into accountspecial repos (repurchase agreements) that render forward replication strategies moreexpensive. Recall from the discussion of the U.S. Treasury market in Chapter 3 thatsupply and demand forces may make a bond scarce, which means that a lender of fundsthrough a reverse repo (buyer of securities to resell tomorrow) will sometimes lendfunds at a zero interest rate just to be able to temporarily own the desired bond. Hisresale price will include a rebate that will make it equal to today’s purchase price, thusearning him no interest.Let us consider a simplified numerical example. Assume that a 1-year zero yielding

6% enjoys general repo while a 2-year zero yielding 6.5% is expected to be on specialfor the next month. You want to replicate a 1-year-by-2-year forward deposit by buyingthe 2-year zero and shorting the 1-year zero. If neither bond were on special, then wecould lock in a rate f12�24 such that:

ð1þ 0:06Þð1þ f12�24Þ ¼ ð1þ 0:065Þ2


Table

6.7

One-yearfloating-rate

bond,quarterly

coupons,face

value:

$200,000

Set

date

Paydate

Daysfrom

Daysfrom

Days

Interpolated

PV

factorto

PV

factorto

Implied

LIB

OR

CF

onpay

PV

ofCF

spotto

spotto

continuous

setdate

paydate

set

date

setdate

paydate

yield

topay

date

25-O

ct-01

25-Jan-02

092

92

2.1728

1.000000

0.994538

2.1489

1,098.34

1,092.34

25-Jan-02

25-A

pr-02

92

182

90

2.1635

0.994538

0.989270

2.1302

1,065.08

1,053.65

25-A

pr-02

25-Jul-02

182

273

91

2.2231

0.989270

0.983510

2.3170

1,171.38

1,152.06

25-Jul-02

25-O

ct-02

273

365

92

2.3314

0.983510

0.976955

2.6252

201,341.78

196,701.94

Sum

ofPVs¼

200,000,00

CF¼

cash

flow

andPV¼

presentvalue.

or f12�24 ¼ 7:0024%. Alternatively, we can write this in terms of discount security pricesas:

1

ð1þ 0:06Þ1

ð1þ f12�24Þ ¼1

ð1þ 0:065Þ2The interpretation of the last equation can be that $1 of principal and interest to bereceived 2 years from today can be guaranteed by spending 1=ð1þ 0:065Þ2 on a 2-yearbond or by spending ½1=ð1þ 0:06Þ�½1=ð1þ f12�24Þ� on a 1-year bond to be rolled overinto 1=ð1þ f12�24Þ of a 1-year-by-2-year forward bond after 1 year. However, in orderto own the 2-year zero we have to give up interest for 1 month. That is equivalent topaying a higher price for the 2-year bond today and is equal to the non-special valuethat we have grossed up plus the lost interest fraction; that is:

1

ð1þ 0:06Þ1

ð1þ f12�24Þ ¼ ð1þ 0:065Þ1=12 1

ð1þ 0:065Þ2The forward that we can really lock in to the market is f12�24 ¼ 6:4423%.When including special repo situations, calculations can get quite complicated,

especially when dealing with coupon securities and trying to compute their forwardprices, which themselves are packages. However, nowhere in this more complicatedanalysis should the principle of arbitrage be violated. We need to be careful toinclude in the mathematics any repo rebates that would have to be given when replicat-ing a forward. The ‘‘specialness’’ of repo cuts both ways: the forward depositor (lender)locks in a lower forward rate; and the borrower does likewise because he uses the hotlydesired security as collateral in a repo transaction. The equations above hold for bothforward lenders and forward borrowers.

6.10 ENERGY FORWARDS AND FUTURES

In addition to crude oil and gas products and their refined derivatives, like gasoline andheating oil, and with the deregulation of electricity markets in the developed world, thelast 5 years have seen the emergence of energy forwards and futures trading for deliveryof electric power to specific points on national grids. Forwards and futures contracts onelectric power inherit certain features from the somewhat complicated spot markets. Inthe U.S., these trade on both a firm and a non-firm basis, depending on whether powerinterruptions are guaranteed by liquidating damages or not. They trade for peak andnon-peak periods which divide each 24-hour period into 16 hours of heavy demand and8 hours of light demand. In Europe, electricity markets trade for base load and peakload periods, the latter being 8 : 00–20 : 00 CET. In addition, geography plays a role asnational grids have natural gridlock points. For example, in the U.S., power to Cali-fornia is delivered through a limited number of access points, like the California–Nevada or California–Arizona borders. Each market may have its own ‘‘convenienceyield’’ issues determined by local deregulation rules concerning power generation andtransmission. In some OTC cases, there are also options embedded in the contractsgiving the short party choices pertaining to the timing of the energy delivery. This canpotentially lead to price manipulation attempts (as claimed in the 2001 Enron scandalsin the U.S.). Most exchange contracts are cash-settled. To prevent settlement price


disputes, exchanges have special provisions for the final futures settlement price. Forexample, EUREX contracts are settled based on the spot prices for base and peak loadsover the entire delivery month. The NYNEX 40-MW contracts for the Pennsylvania–Maryland–New Jersey hub settle based on the arithmetical average of prices over thepeak days during the delivery month. The COB (California–Oregon border) and PV(Palo Verde, Arizona) contracts call for 432MWh delivered at the rate of 1MWh perhour during 16 peak hours spread over 27 non-Sunday days. While such tight standard-ization is desirable for liquidity and price stability, it often means that the futurescontracts may not be the best hedge instruments for power generators and transmitterswho face specific local conditions. That is, perhaps, why the market is dominated bycustomized OTC contracts. These are arranged through a small number of players whoact almost like exchanges (before its collapse, Enron established itself as such anexchange).

Studies show that futures prices do in fact converge to the spot prices for the deliverypoints on the grid by the expiry month. That is, at least one part of the cost-of-carryequation is satisfied: the financing cost; this, inclusive of the convenience yield, goesdown to zero. But studies also show that hedging with futures does not reduce thevolatility of the average price paid, especially if hedgers take basis risk between theactual delivery point and time, and the contract specifications. This is because both spotand futures prices are highly variable at all times. Power suppliers can only reduce thisrisk by trading off the stability of long-term contracts for higher overall cost, supple-mented with some futures hedging.


___________________________________________________________________________________________________________________________________________________________________________ 7 __________________________________________________________________________________________________________________________________________________________________________

________________________________________________________________________________ Spot–Forward Arbitrage _______________________________________________________________________________

The cash-and-carry trade is arguably the most important principle linking spot andforward markets for all securities. Its essence lies in going long a synthetic forwardand short a real forward, or short a synthetic forward and long a real forward. Thesynthetic side of the trade relies on cash flow replication through a spot positioncombined with borrowing or lending. A synthetic long forward is constructed using aspot purchase and a spot borrowing transaction. A synthetic short forward is con-structed using a spot sale and a spot lending transaction. For example, a forwardpurchase of a bond is replicated by borrowing money to buy the bond now andrepaying the borrowing at a future date. A forward sale of a stock is replicated byselling the stock now and investing the proceeds at some interest rate. A forwardcommodity purchase is constructed by borrowing the money to buy the commoditynow and paying for storing it between now and a future date. The synthetic side, whichmimics the forward completely, is traded against the forward; the sides are chosenbased on the relative costs of the two. The trade is viable until the costs are the sameand money cannot be made; that is, when the markets adjust to follow the mathe-matical cost-of-carry formulae we have laid out: spot and forward foreign exchange(FX) rates fall in line with (borrowing and lending) markets; spot stock indexes(baskets) and index futures adjust to rates for borrowing and lending against equitycollateral; coupon bonds, zeros, forward bond sales and purchases, and forward depositrates fall in line with spot and repo (repurchase agreement) rates.The cash-and-carry building blocks can also be combined to perform forward–

forward arbitrage where the cost-of-carry is locked in today, but for forward periods.This can be viewed as two cash-and-carry transactions in opposite directions combinedin one package (with spots canceling out). The forward–forward mechanism strength-ens the arbitrage relationships between various markets as the forward replication isperformed using not just spots, but also other forwards, and the spot replication can beperformed by using an entire menu of forwards on the same underlying cash instru-ment. For example, stock index arbitrage cannot only tie the spot value of the index tothe index futures price through borrowing/lending markets, but also index futures ofdifferent maturities to each other through forward borrowing/lending transactions,executed in the forward rate agreement (FRA) or Eurocurrency markets.In this chapter, we use the mathematics of Chapter 6 to identify cash-and-carry

arbitrage in a variety of markets. We start with currencies where the syntheticforward always combines spot with both borrowing and lending. We cover a simplercase of stock index arbitrage where the synthetic forward combines spot with eitherborrowing or lending, and the other part is replaced by dividends. We describe a similarcase of bond futures arbitrage where dividends are replaced by coupons. We alsoforward trades that link zeros and coupons. All four of these trades are benchmarkcases in their own markets. Relative value trades deviate from these pure arbitrage cases

due to imperfect synthetic replication or basis risk on one side of the trade. We devotetwo short sections to dynamic hedging with a Eurodollar strip and via duration match-ing. The latter are examples of imperfect replication through continuous rebalancingand as such are relative value strategies. In those cases, a series of risks (e.g., exposureto many points on the yield curve) is reduced to its main components (e.g., only the2-year and the 10-year point) and hedged as such. This is expedient and involves smallertransaction costs, but requires rebalancing over time and leads to a tracking error. It isthe most common method for hedging large fixed income portfolios.

7.1 CURRENCY ARBITRAGE

To review the spot–forward currency arbitrage based on the covered interest rate parity(CIRP) principle, let us use the data for USD and EUR as of August 13, 2003.1 Hereare our quotes.

FX rates [USD/EUR] FX (CME) futures (settle)Bid/Ask Mid

Spot 1.1308/1.1315 1.131 151m (18-Sep) 1.1296/1.1307 1.130 05 Sept (19-Sep) 1.13002m (20-Oct) 1.1285/1.1297 1.129 13m (18-Nov) 1.1276/1.1288 1.128 24m (18-Dec) 1.1267/1.1279 1.127 3 Dec (19-Dec) 1.12725m (20-Jan) 1.1257/1.1269 1.126 36m (18-Feb) 1.1249/1.1261 1.125 5 Mr04 (19-Mar) 1.12489m (18-May) 1.1224/1.1236 1.123 01y (18-Aug) 1.1202/1.1214 1.120 82y (18-Aug) 1.1172/1.1193 1.118 25

We also have the following LIBOR (London interbank offered rate) (ask) rates:

Dollar (UBS&WSJ) Euro (UBS) Euro (WSJ)1-month 1.100 0% 2.144 76% 2.115 382-month 1.120 0% 2.157 43%3-month 1.130 0% 2.162 12% 2.132 504-month 1.148 75% 2.169 98%5-month 1.168 75% 2.182 4%6-month 1.180 0% 2.188 73% 2.158 7512-month 1.36% 2.281 64% 2.250 38

Let us just check a few CIRP equations of the form:

FN-month

�USD

EUR

�¼ S

�USD

EUR

��1þ LN-month

USD

Act0�N

360

1þ LN-monthEUR

Act0�N

360


1 The source is the August 14, 2003 issue of the Wall Street Journal and a UBS website: http://quotes.ubs.com/quotes/Language=E accessed on August 13, 2003.

where LN-monthCURR is an N-month LIBOR for currency CURR, and Act0�N is the number

of days between month 0 (i.e., today) and the end of month N, which is the maturity ofthe forward for the implied borrowing/lending transactions. For FX forwards with 3-and 6-month expiries, we get:

F 3m ¼ 1:131 151þ 0:011 300� 97

360

1þ 0:021 6212� 97

360

¼ 1:128 023

F 6m ¼ 1:131 151þ 0:011 800� 189

360

1þ 0:021 8873� 189

360

¼ 1:125 228

Ideally, we would want to write two separate equations for each maturity: one assuminga spot bid for euros, a lending rate for euros (i.e., London interbank bid rate or LIBID,not LIBOR), and a borrowing rate for dollars (i.e., a LIBOR which is ask); and anotherassuming a spot ask for euros, a borrowing rate for euros (LIBOR) and a lending ratefor dollars (LIBID), as these would precisely reflect our synthetic forwards. This wouldproduce bid and ask quotes for the synthesized forwards and would define a no-arbitrage zone. For simplicity, we assume that we can trade at mid-rates (averages ofbids and asks), whether buying or selling, or borrowing or lending.The calculated fair value forward rates fall within the actual forward quotes, reflect-

ing no arbitrage. Now suppose that the dealer changes the quotes for 3- and 6-monthforwards to mid-rates of:

Spot 1.131 15 (Size: EUR 100,000,000)3m 1.129 7 (Size: EUR 50,000,000)6m 1.124 0 (Size: EUR 50,000,000)

Note that for the 3-month quote the fair value of 1.128 023 is well below the actualquote of 1.1297. That means that in the forward market euros are rich (expensiverelative to synthetic). This sets the direction of our arbitrage: we will want to selleuros forward. To match the flows, we use circular thinking. In order to sell euros 3months forward, we need to deposit them today to earn interest. In order to depositthem today, we need to buy them spot by selling dollars. In order to sell dollars spot, weneed to borrow them. If we borrow dollars today, we will have to repay the principaland interest 3 months from today. By selling euros forward, we will obtain dollars torepay the borrowing. That completes the circle. Now all we have to do is to compute theamounts in such a way that we have net zero cash flows in both currencies 3 monthsfrom today and such that we have a positive cash flow in one or both currencies today.Suppose we are a New York dealer and want to earn profit in dollars. We compute theamounts by working backwards from the forward transaction whose size is limited bythe quote size of EUR 50,000,000 (we would like that size to be unlimited to make themost money!). Today, we enter into the following contracts:

. Sell EUR 50,000,000 3-month forward at EUR/USD 1.1297 for USD 56,485,000.

Spot–Forward Arbitrage 177

. Deposit (lend) EUR 49,710,401.51 at 2.16212% for 3 months, since

50,000,000=(1þ 0:021 621 2� 97=360) ¼ 49,710,401:51

. Borrow USD 56,313,540.91 at 1.13% for 3 months, since

56,485,000=(1þ 0:0113� 97=360) ¼ 56,313,540:91

. Sell USD 56,229,920.66 spot at USD/EUR 1.13115 to buy EUR 49,710.401.51 inorder to cover the EUR deposit.

In 3 months’ time, we have zero net cash flows in both currencies:

. The EUR deposit accrues with interest to EUR 50,000,000 and is withdrawn todeliver to the forward contract.

. The USD 56,485,000 obtained from the forward contract is used to pay off theaccrued loan liability of exactly that amount.

Today, we have a zero net cash flow in euros, but a positive cash flow in dollars:

. The EUR 49,710,401.51 purchased spot is deposited for 3 months.

. We borrow USD 56,313,540.91, but sell spot for euros only which gives us USD56,229,920.66, resulting in a profit of USD 83,620.25.

The flow of the transactions is portrayed in Figure 7.1 (þ is a cash inflow, � is anoutflow).

Figure 7.1 Spot–Forward currency arbitrage based on covered interest-rate parity for 3 months.

Note that a dealer based in Europe would have spot-exchanged all of the borrowedUSD 56,313,540.91 into EUR 49,784,326.49 and lent out the same EUR 49,710,401.51,resulting in a net profit of EUR 73,924.98. Also note that the fact that we are lending inEUR with a higher interest rate and borrowing in USD with a lower interest rate iscoincidental here and in and of itself did not determine the direction of the transactions.It was strictly the comparison of the fair value forward to the actual forward. The nextexample will make that point clear.

Let us now create profit out of the 6-month quote. The fair value of 1.125 228 is well


above the actual quote of 1.1240. That means that in the forward market euros arecheap relative to the synthetic strategy. This time we will want to buy euros forward. Inorder to sell dollars 6 months forward, we need to deposit them today to earn interest.In order to deposit them today, we need to buy them spot by selling euros. In order tosell euros spot, we need to borrow them. If we borrow euros today, we will have torepay the principal and interest 6 months from today. By selling dollars forward, wewill obtain euros to repay the borrowing. Again we work backwards from the FXforward to have net zero cash flows in both currencies 6 months from today and tohave a positive cash flow in one or both currencies today.The flow of the transactions for a New York dealer locking in profit in dollars is

portrayed in Figure 7.2 (þ is a cash inflow, � is an outflow).

Figure 7.2 Spot–Forward currency arbitrage based on covered interest-rate parity for 6 months.

The borrowing and lending amounts are computed, like in the 3-month example, bytaking the present values of the forward amounts at the respective interest rates, i.e.:

50,000,000=(1þ 0:021 887 3� 189=360) ¼ 49,431,985:34

56,200,000=(1þ 0:011 800 0� 189=360) ¼ 55,853,984:57

The spot FX transaction results in dollars in excess of the amount needed to deposit,resulting in a profit of USD 61,005.65. (Here a European dealer would have spot-exchanged a smaller amount of euros needed to generate exactly USD 55,853,984.57required for deposit, leaving himself with a profit in euros.)Note that in this example the arbitrageur borrows in the high interest rate currency,

the euro, and lends in the low interest rate currency, the dollar. Although this may seemcounterintuitive, this yield loss is already taken into account in the fair value equationswhich consider the total cost of synthesizing forward dollars and euros. Based on that,euros turn out to be expensive and dollars cheap. The dealer synthesizes an outflow ofeuros and an inflow of dollars (i.e., a forward sale of euros for dollars), and offsets theseflows with a transaction in the forward market in the opposite direction. In the processof creating the synthetic forward, he borrows in euros and lends dollars, locking in ayield loss. He is more than compensated for that loss by the difference between the


actual and the synthetic forward. Going in the other direction would lock in a risklessloss of the same amount.

Note also that both American and European dealers would perform the same trans-actions, selling euros spot and buying them forward, as well as borrowing euros andlending dollars for 6 months. All would exert pressure on the FX and interest rates tomove into parity with each other. The very attempt to lock in profit may eliminate itspossibility.

Let us now illustrate forward–forward arbitrage with FX rates. Suppose that thereare no arbitrage opportunities in the interest rate market. That is, based on our spotLIBOR quotes, 3� 6 FRAs are quoted 1.228 976% for U.S. dollars and 2.203 947% foreuros. We obtain these numbers by solving for f3�6 in the following fair value equationfor interest rates applied to both currencies:�

1þ L3mAct0�3

360

��1þ f3�6

Act3�6

360

�¼ 1þ L6m Act0�6

360

For dollars, we have:

f3�6 ¼ 360

92

1þ 0:0118� 189

360

1þ 0:0113� 97

360

� 1

0B@

1CA ¼ 0:012 289 76

and for euros we have:

f3�6 ¼ 360

92

1þ 0:021 587 5� 189

360

1þ 0:021 325 0� 97

360

� 1

0B@

1CA ¼ 0:022 039 47

We assume that we can lock in borrowing and lending rates in both currencies at theselevels. Suppose also that we get quotes for 3- and 6-month FX forwards:

Spot 1.131 15 (Size: EUR 100,000,000)3m 1.129 7 (Size: EUR 50,000,000)6m 1.124 0 (Size: EUR 50,000,000)

We have already computed fair value forwards using the CIRP link to the spot. Thesewere 1.128 023 and 1.125 228. We can also write a 3-month-by-6-month forward CIRPlink with n ¼ 3 and N ¼ 6 in the following way:

FN-month

�USD

EUR

�¼ Fn-month

�USD

EUR

��

1þ f USDn�N

Actn�N

360

1þ f EURn�N

Actn�N

360

We can synthesize the 6-month FX forward with a strategy involving a 3-month FXforward and forward borrowing/lending in the two currencies. Relative to the quoted 3-month FX forward, the fair value of the 6-month forward should be:

F6m ¼ 1:1297�1þ 0:012 289 76

92

360

1þ 0:022 039 4792

360

¼ 1:126 901


(i.e., much higher than the actual 6-month forward of 1.1240). The price of the euro indollars for the delivery in 6 months synthesized from a 3-month forward and deposittransactions is higher than the actual forward price. To profit, we will buy eurosthrough the actual 6-month forward and sell them forward synthetically. Like before,this sets the circular direction of all transactions.To buy euros (sell dollars) 6 months forward, we will need to deposit dollars 3

months from today for 3 months. To supply dollars to the deposit, we will buydollars by selling euros 3 months forward. We will borrow euros forward 3 monthsfrom today for 3 months. We will earn interest on the deposited dollars and pay intereston the borrowed euros. We can lock in those rates in today’s FRA markets (or furthersynthesize the forward loan in each currency by two offsetting spot deposit transactionsfor 3 and 6 months). All the contracts are entered into today, and the profit is locked in3 months from today.

Figure 7.3 Forward–Forward currency arbitrage based on covered interest-rate parity in 3months for 3 months.

Against the anticipated profit of USD 144,596.48, we can borrow today in the Euro-dollar market the amount 144,596:48=(1þ 0:011 300� 97=360) ¼ 144,157:56. Thisborrowing locks in a profit of USD 144,157.56 today.In Figure 7.3, we lend dollars forward at 1.228 976% and borrow euros forward at

2.203 947%. In reality, we will lend dollars in 3 months at the then-prevailing 3-monthLIBOR rate, but we lock in the net interest rate by contracting today to receivefixed 1.228 976% on a 3� 6 dollar FRA. Similarly, we will borrow euros at the then-prevailing LIBOR rate, but we lock in the net interest rate by contracting today to payfixed 2.203 947% on a 3� 6 euro FRA. Each FRA is written for the principal amountcomputed above as the borrowing/lending amount 3 months from today.As before, all dealers will enter into the same trades causing the FX and money rates

to fall in line. The spot–forward and the forward–forward arbitrage will act in the samedirection, further strengthening the CIRP relationships (and eliminating the very arbi-trage opportunities).


7.2 STOCK INDEX ARBITRAGE AND PROGRAM TRADING

Stock index arbitrage follows the same cost-of-carry logic as CIRP-based currencyarbitrage. The only difference is that the role of foreign currency lending/borrowingis played here by the anticipated stock dividends. To profit from a situation of futuresprice exceeding its fair value we short actual futures contracts and synthesize a longfutures position: we borrow cash by agreeing to pay interest, we buy a basket of stocksto mimic the performance of the index, and we earn dividends on holding the basket tothe maturity date. We match the amounts involved in such a way that we have zero netcash flows on the futures expiry date and a positive cash flow today. To profit from asituation of futures price being below its fair value, we reverse the strategy. We buyactual futures contracts and synthesize a short futures position. We short an index-mimicking basket by borrowing the physical assets (i.e., stocks), we invest the proceedsfrom the short sale at an interest, and we reimburse the lender of the stocks for the cashdividends until the maturity date.

We will illustrate this using the same July 21, 2003 data as in Chapter 6. Let us recallthat with the S&P 500 index equal to 978.80 at the close of the day, the futuressettlement prices were:

S&P 500 index (CME)—$250� indexSep03 978.00Dec03 976.10Mar04 974.20

and the U.S. dollar LIBOR rates stood at:

1-month 1.100 0%3-month 1.110 0%6-month 1.120 0%12-month 1.211 25%

We consider the September contract with 2 months left to expiry. We use an inter-polated LIBOR rate of 1.105% and 60 as the number of days left to the September 19expiry date. We also assume that if we were to spend exactly $978.80 to buy the 500stocks represented in the index in the proportions as defined by the index (admittedlythis would mean buying very tiny amounts of each stock, but we will deal with muchlarger dollar amounts so that everything will scale up to realistic numbers), then, overthe holding period between today and September 19, we would accumulate dividendsfrom the constituent stocks in the amount of D ¼ $2.50 as of September 19.2 We cancompute the fair value of the futures to reflect the cash-and-carry strategy as:

F ¼ S � D

1þ L� Act

360

0B@

1CA�

�1þ L� Act

360

�


2 Almost all stock indices around the world are based purely on stock prices and exclude dividends. Our treatment assumesthat this is the case and we treat dividends as ‘‘interest’’ earned on holding stocks. The DAX index is an exception asdividends are assumed to be reinvested and are thus included in the index itself.

and specifically in our case:

F ¼ 978:80� 2:50

1þ 0:011 05� 60

360

0B@

1CA�

�1þ 0:011 05� 60

360

�¼ 978:1026

Analogously to investing in a LIBOR deposit in a foreign currency, we can think of the$2.50 total dividend as ‘‘interest’’ earned on holding stock. Note that the $2.50 is aprecise estimate based on the schedule of dividend payments announced by eachcompany in the index for the period from July 21 to September 19. This numbermight vary throughout the year as each company would follow its own calendar ofex-dividend dates. These may or may not coincide with the calendar year. Over longerholding horizons, say a few years, we may be less confident about our future dividendestimates. In that case, it is more common to be certain about dividend yields thandollar amounts (dollar amounts are likely to grow over many years as stock pricesgrow). To illustrate working with dividend yields, we can convert the $2.50 amount to ayield d on a discount instrument with a face value equal to the current price of the share(i.e., $978.80) sold today for the current price of the share, $978.80, minus the presentvalue of the anticipated dividends. Dividend yield d can be defined on any day-count orcompounding basis, including continuous. Here we use an Act/365 basis. The dividendyield d must thus satisfy the following equation:

S � D

1þ L� Act

360

0B@

1CA�

�1þ d � Act

365

�¼ S

or specifically in our case:

978:80� 2:50

1þ 0:011 05� 60

360

0B@

1CA�

�1þ d � 60

365

�¼ 978:80

Solving, we get d ¼ 1:554 881%. By defining the dividend yield in this manner, we canrewrite the fair value equation for stock and stock index futures to look analogous tothat based on CIRP for currencies:

F ¼ S1þ L� Act

360

1þ d � Act

365

Substituting the numbers we get as before:

F ¼ 978:801þ 0:011 050 00� 60

360

1þ 0:015 548 81� 60

365

¼ 978:1026

The actual September futures price is 978.00. Comparing the fair value to the actualprice, we can make the following statement. The S&P 500 basket can be bought fordelivery in September for $978.00 by going long September futures; this is cheaper than


by synthesizing the purchase with the use of spot equity markets and the moneymarkets for borrowing against stock collateral. In order to profit, we will want tobuy stocks forward (i.e., by going long in the futures and sell stock through a syntheticfutures contract). To maximize our profit, we would want to buy as many futurescontracts as possible at the ‘‘unfairly low’’ price of 978.00 and short as many basketsas possible at the ‘‘unfairly rich’’ synthetic forward price of the basket of 978.1026,based on the spot index value of 978.80. By attempting this strategy, we would mostlikely drive the futures price up and the spot index basket price down to bring the two inline with the cost-of-carry equation. With shorting 500 stocks we would inevitablyencounter thin markets in some of the component stocks so that our realized indexvalue (average actual sale price of the basket) may already be lower than the publishedindex value. This is why the speed of execution and size of orders sent to the floor of theexchange matter so much for spot–futures arbitrageurs.

Let us now determine the amounts of the transactions for one futures contract with amultiplier of $250 per index point. As with currencies, we will work backwards bymatching future cash amounts and stock flows first, leaving ourselves with profit today:

. Go long one futures contract to lock in a price of buying 250 stock baskets forwardfor

250� 978:00 ¼ $244,500

. We reimburse the lender of the stocks for lost dividends on the forward date

250� 2:50 ¼ $625

. The total (principal and interest) we need to receive from a deposit on a forward dateis thus

244,500þ 625 ¼ $245,125

. This means we can lend (invest) today

245,125=ð1þ 0:011 05� 60=360Þ ¼ $244,674:39

. We short-sell 250 index baskets at $978.80 each for $244,700.

This leaves us with a zero net cash and stock flow on the forward date (futures expiry):

. We buy the stocks back at the then-current spot price (say $980.35), but we have netvariation margin flow (of 980:35� 978:00¼ $2.35 per basket); our net cost of acquir-ing the baskets is exactly the original futures price of $978.00 times the number ofbaskets, 250, or $244,500. This, together with the dividend reimbursement of $625, ispaid for in full by the maturing LIBOR deposit.

. The stocks purchased in the open market are returned to the lender of the stock.

Today, we have a positive cash flow of $25.61, equal to the excess of our proceeds fromthe short sale over the amount we lend to the LIBOR deposit. Stocks are borrowed andsold (short) in the market leaving us with no net stock flow.

We summarize the amounts in the now-familiar Figure 7.4:


Figure 7.4 Stock index futures arbitrage.

Let us now show an arbitrage strategy for the situation where the fair price is lowerthan the futures price. Suppose that on July 21, 2003 the September S&P 500 futuressettle at 978.30 and we are able to short contracts right at that price. Since the fair valueof 978.1026 is lower than the actual of 978.30, we would want to go short futures andlong synthetic forwards. We compute the amounts backwards by matching flows:

. Go short one futures contract to lock in a price of buying 250 stock baskets forwardfor

250� 978:30 ¼ $244,575

. We will receive dividends on the forward date

250� 2:50 ¼ $625

. Total (principal and interest) that we can repay on an interest deposit on a forwarddate is thus

244,575þ 625 ¼ $245; 200

. This means we can borrow today

245,200=ð1þ 0:011 05� 60=360Þ ¼ $244,749:25

. We buy 250 index baskets at $978.80 each for $244,700.

This leaves us with a zero net cash and stock flow on the forward date (futuresexpiry):

. We sell the stocks held at the then-current spot price (say, $980.35), but we have netvariation margin flow of (�1)� (980.35� 978.00)¼ $�2:35 per basket. Our net costof selling the baskets is exactly the original contracted futures price of $978.00 timesthe number of baskets, 250, or the total of $244,500. This, together with the receiveddividends of $625, is used to pay off the maturing LIBOR borrowing.

Today, we have a positive cash flow of $49.25, equal to the excess of LIBOR borrowingover the purchase price of the stocks. Stocks are purchased and deposited in ouraccount to earn dividends, leaving us with no net stock flow.Again we summarize the amounts in a diagram (Figure 7.5):


Figure 7.5 Stock index futures arbitrage with dividends.

Note that the above discussion is simpler in the case of individual stock futures, likethose on eBay’s shares. We do not need to trade a whole basket of stocks, but simply acertain number of shares of the same stock (100 shares of eBay per one futures con-tract). The borrowing/lending amounts are arrived at analogously to the index case,working backwards from the forward amounts.

Stock index arbitrage is primarily executed with the use of program trading. A dealercontinuously computes the fair value of futures, adjusting it for his estimated executioncost. The latter is related to the speed of execution, the depth of the market in eachconstituent stock, and the dealer’s own settlement costs. If the program is slow, therealized price of each basket traded may be different from the one assumed in the fairvalue calculation. The dealer’s estimated profit must exceed all these costs. A dealermay at times resort to mildly speculative tricks of the trade, like sending some orders asmarket and some as limit, front-running the anticipated moves in the futures price, etc.,in order to beat his costs. A dealer may also take on what is called a tracking, orcorrelation risk. Instead of sending a program order for all stocks in the index (500for S&P 500), he may trade in only a subset of the stocks. This can be justified on twogrounds. Defensively, as some stocks may have thinner markets (fewer bids and offers)and the price concessions or markups when trying to replicate the index may be large.Offensively, the dealer, based on fundamental research, deletes stocks from the programbelieving that his imperfect basket will outperform the prescribed one. For example, if astock is rated ‘‘sell’’, the dealer may want to exclude it from a buy program. Exclusionsmay also be driven by external factors, like regulation. A broker-dealer engaged in aninvestment banking transaction (bond or stock issue) with a particular company may beprohibited from transacting in the company’s stock (the stock may be placed ‘‘on therestricted list’’ to prevent new issue price manipulation).

Program trades can also be netted. If an SPX transaction calls for spot basket sales,but a Dow futures strategy calls for spot basket purchases, the two can be netted toyield a net basket sale. This may happen especially when the dealer takes on a trackingerror risk.

The risks of program trading—stock index arbitrage and customer programexecution—are related to external execution risks and deliberate risk taking. External


execution risks are always present and are amplified during large market dislocations,like market crashes. When stocks fall rapidly, it is much easier to sell futures than to sell500 different stocks, some of which may trade on different exchanges. At that moment,the futures price is a more reliable determinant of the spot fair value than vice versa.This is also true for after-hours trading (e.g., right before cash markets open). TVcommentators make predictions for the stock market openings based on this reversedfair value calculation, by knowing the current futures price. For example, at 9 : 20 ESTin New York, the U.S. stock markets are closed but the futures are not. By computingthe fair value of spot, using the reverse cash-and-carry argument, we can make aprediction about the stock market opening 10 minutes later.Deliberate risk taking by assuming a tracking error is a form of relative value trading,

or quasi-arbitrage. This case should be viewed as two separate strategies, futuresagainst the correct basket and the correct basket against the simplified trackingbasket. The former should result in sure profit, the latter can be a source of profit orloss. When performing index arbitrage or customer program execution, dealers competewith each other not only on pure execution costs, but also on this extra ‘‘skill’’ ofoutperforming the market. Bids for customer programs submitted by large dealersoften reflect their perceived advantage in a market segment in which the program isconcentrated. The dealer may be a large-volume market-maker or possess superiorexecution technology (high speed of execution and error control).

7.3 BOND FUTURES ARBITRAGE

Bond futures arbitrage is a classic example of a cash-and-carry trade described inprofessional fixed income textbooks. Yet it is quite a bit more complicated than itscurrency and stock index counterparts. There are two reasons for this. First, thecomputation of the forward price of the bond relies on the reinvestment rate for theintermediate coupon accrual or receipt (between now and futures maturity) and, moreimportantly, the repo rate for the bond. And since the repo market tends to be veryshort-term (overnight or a few days), the actual repo interest paid/received on lending/borrowing the bond may not be known in advance unless we can negotiate a term repoto the futures maturity. Second, most bond futures have delivery and timing optionsembedded in them which afford the short-futures party the right to choose a bond to bedelivered from a long list of eligible bonds and the timing of the settlement. Theseoptions may result in a mismatch between the bond chosen for the synthetic cash-and-carry strategy and the actual bond delivered to the futures contract, resulting inan unknown profit or loss at expiry.Let us take a close look at the U.S. long-bond contract traded on the CBT (Chicago

Board of Trade). In order to prevent a squeeze on any one bond, the contract specifiesthat the deliverer of the bonds on the expiry date may choose from a list of eligiblebonds of similar maturity (minimum 15 years). The long party receives a bond with aface value of $100,000 and pays an invoice price equal to the final futures settlementprice times a conversion factor3 for the bond delivered (plus accrued interest), just as if


3 They are called price factors on LIFFE.

the quantity of the delivered bond were different. The conversion factor is the percen-tage price of the bond computed at an artificial yield of 6%. (A bond priced at 6% to10916

32has a conversion factor of 1.0950.) This is meant to make all bonds close sub-

stitutes of an artificial 6% par bond and of each other (i.e., to have the same value). Inreality it does not, making some bonds more likely to be delivered (i.e., cheaper) thanothers.4

The most common, albeit not the best, way to determine which of the eligible bondsis the cheapest-to-deliver (CTD) is to compute each bond’s basis. The basis for a givenbond is the difference between the actual price of the bond in the market and the futuresprice times the conversion factor for that bond:

Basis in ticks ¼ (Bond price� Futures price� Conversion factor)� 32

The higher the basis, the costlier it is for the short-futures party to purchase the bond inthe market and sell it through futures for the settlement price times the factor. A betterway to determine the CTD is to find the highest repo rate implied in purchasing a bondand selling futures.5 The amount of futures sold is equal to the conversion factor for thecash-and-carry bond. The cash-futures trade is equivalent to lending cash/borrowingthe bond in the repo market.

Once we have decided on the CTD, the mechanics of the bond–futures arbitrage areanalogous to stock futures. Similar to dividends received between the spot date and thefutures expiry date, we have to account for the coupon accrual between those two dates.The fair value of the futures can be written as:

Ffair ¼ 1

CvFactor��B�

�1þ r� Act

360

�� FV(Accr)

�

where the future value of the bond’s spot price B is taken using the repo rate r.We subtract the future value of the accrual, Accr, to reflect the portion of thecoupon accrued by the bond holder between now and expiry (i.e., the differencebetween the accrued interest on the expiry date and accrued interest on the spotdate). The treatment varies if there is an actual coupon received in the interimperiod. The conversion factor, CvFactor, scales the number of futures contracts inthe cost-of-carry relationship.

We will dispense with an actual numerical example here only to state that the cash-and-carry arbitrage can take two forms. If the fair value is lower than the actual futuresprice, we go long the basis (i.e., purchase a deliverable cash bond, financing it in the repomarket, and sell a factor-weighted number of futures). If the fair value exceeds thefutures price, we go short the basis, by shorting the bond, receiving the repo interest,and buying a factor-weighted number of futures contracts. We match the amounts forthe expiry date. Without using numbers, we can summarize the long-basis trade inFigure 7.6:


4 For a good discussion of factors and delivery issues, see an older text by Daniel R. Siegel and Diane F. Siegel, The FuturesMarkets, 1990, Probus Publishing, Chicago.5 Bloomberg screens can order bonds from the highest to the lowest implied repo rate (i.e., from the most likely to be the CTDto the least likely).

Figure 7.6 Bond futures arbitrage.

7.4. SPOT–FORWARD ARBITRAGE

IN FIXED INCOME MARKETS

In Chapter 6, we described the linkages between the zero and coupon instruments of thespot market and the forwards and futures on deposit rates. We covered the syntheticreplication of spots from forwards and vice versa as well as the bootstrap of the zero-coupon curve from forwards. In addition, in Chapter 2, we covered the bootstrap of thezero-coupon curve from the spot coupon bonds. If any of the synthetic structuresdescribed in these linkages do not result in the same exact financing rates as theiractual counterparts, temporary riskless profit opportunities arise.

Zero–Forward trades

Suppose we observe the following set of zero rates and a set of FRA rates (Table 7.1) ona given day, all on a 30/360 basis.

Table 7.1 Spot and forward zero-coupon rates

Zero rates FRAs(semi except *) (semi)

3 month* 2.454 476 month 2.500 01 year 2.520 0 6� 12 2.540018 month 2.550 0 12� 18 2.61002 year 2.575 0 18� 24 2.603030 month 2.594 6 24� 30 2.72003 year 2.621 3 30� 36 2.7550


We use the familiar no-arbitrage condition for all successive zero rates (ignoring the3-month rate): �

1þ zn

2

�n=6

¼�1þ zn�6

2

�n=6�1

�1þ fn�1�n

2

�

where zn is a zero rate with a maturity of n months and fn�1�n is an FRA rate with astart maturity of n� 1 months and end maturity of n months. For example, for the 18-month zero, we use:�

1þ z18

2

�3

¼�1þ z12

2

�2�1þ f12�18

2

�¼

�1þ 0:0252

2

�2�1þ 0:0261

2

�

resulting in a fair 18-month zero of z�18 ¼ 2:55%. This happens to be equal to the actual18-month zero of 2.55%. But when we apply the equation one more time to computethe fair 2-year zero, we get z�24 ¼ 2:5632% instead of the actual 2-year zero of 2.575%.We have discovered an arbitrage opportunity. We cannot claim that the 2-year zero is‘‘unfair’’ or ‘‘too high’’. We could just as easily claim that the 1-year zero and the12� 18 FRA are ‘‘too low’’, producing a low synthetic rate. We are simply observingthat the 2-year zero can be manufactured from other instruments, or synthesized, toyield less than the actual 2-year zero-coupon bond.

In our example, we have only one ‘‘misquote’’, and it happens to be the 2-year zero.This can be verified by checking alternative ways of synthesizing zeros. The 30-monthzero rate could be synthesized from the 2-year zero and the 24� 30 FRA. It can also besynthesized from a 1-year zero, 12� 18, 18� 24, and 24� 30 FRA, or any othercombination of lower maturity zeros and forwards. By performing the calculationsfor all these combinations, we will discover that the only one that produces a ratenot equal to the actual is the one involving the 2-year zero. All the others will yield afair 30-month zero rate of exactly 2.5946%. The same is true for the 36-month zerorate.

We can structure several distinct arbitrage strategies. All strategies will involvelending at the 2-year rate of 2.575% and synthetic borrowing through a combinationof zeros and forwards. One example is to borrow for 3 years at 2.6213% and shortenthe borrowing maturity to 2 years, by receiving on the 24� 30 FRA at 2.72% and onthe 30� 36 FRA at 2.755%. The simplest strategy is to:

. Lend for 24 months by buying a 2-year zero yielding 2.575%.

. Borrow for 18 months by selling an 18-month zero yielding 2.55%.

. Extend the borrowing for another 6 months by agreeing to pay 2.603% on an18� 24 FRA.

We match the amounts for future dates, leaving us with a borrowed amount in excess ofthe lent amount as riskless profit. The limiting factor is the depth of the quotes. Supposethat at most we can buy $100 million face value of the 2-year zero. We compute the restof the numbers working backwards from $100 million:

. The notional principal on the 18� 24 FRA will be 100,000,000/(1þ 0.026 03/2) =$98,715,221.39.

. That will also be the face value of the 18-month zero for which we will receive

98,715,221:39=(1þ 0:0255=2)3 ¼ $95,033,648:12


. The 2-year zero with a $100 million face value, yielding 2.575%, will cost us

100,000,000=ð1þ 0:025 75=2Þ4 ¼ $95,011,591:39

All cash flows in the future are matched. When the 18-month borrowing of$98,715,221.39 matures we will pay it off by borrowing that amount in the 6-monthLIBOR market at the then-prevailing rate. However, by signing the 18� 24 FRA today(to pay fixed) we are guaranteeing the effective rate on that borrowing (our net costincluding the FRA settlement amount) to be 2.603%. We will thus owe exactly $100million in 2 years. That is exactly the face value of the 2-year zero-coupon we areinvesting in today. Our riskless profit from the strategy is $22,056.73. The strategy isa disguised cash-and-carry trade where we buy a 2-year bond in the cash market andcarry (finance) it by borrowing for 18 months. Against that we sell forward bonds byagreeing to pay fixed. The same strategy executed with Eurodollar (ED) futures on 6-month LIBOR, instead of FRAs, would involve selling 98.715 contracts at 97.397 (i.e.,100.000� 2.603). We discuss in the next sections how the actual ED futures on 3-monthLIBOR are used in locking rates.

Coupon–Forward trades

To the extent that coupon bonds are packages of zero-coupon bonds and we have justshown that zero-coupon bonds are packages of forwards, we can construct a two-layerstrategy involving coupon bonds traded against forwards. The arbitrage can arise whencoupons are mispriced relative to zeros, but we prefer to use forwards or futures, orwhen coupons are mispriced relative to forwards. We discover the latter when weobserve par coupon rates, construct the discount curve, and discover that the latterdoes not agree with the forwards and futures. Let us look at this case more closely.Suppose that we obtain the following quotes (Table 7.2), all semi-annual on a 30/360

basis:

Table 7.2 Spot coupon and forward zero rates

Coupon rates FRAs(semi) (semi)

6 month 2.50001 year 2.5199 6� 12 2.540018 month 2.5495 12� 18 2.61002 year 2.5628 18� 24 2.6030

Using the par coupon rates, we bootstrap the semi-annual zero-coupon (discount curve)as described in Chapter 2. That is, the subsequent zero rates are backed out using thefollowing no-arbitrage equation:

cn=2

(1þ z6=2)þ cn=2

(1þ z12=2)2þ � � � þ cn=2

(1þ zn=2)n=6þ 100

(1þ zn=2)n=6¼ 100

with the first zero with a 6-month maturity being automatically equal to the first parrate (i.e., Z6 ¼ 2:50% (( c6 ¼ 2:50)). For example, the 12-month zero is obtained


from:

cn=2

(1þ z6=2)þ cn=2

(1þ z12=2)2þ 100

(1þ z12=2)2¼ 2:5199=2

(1þ 0:0250=2)þ 2:5199=2

(1þ z12=2)2þ 100

(1þ z12=2)2¼100

and so on. By following this procedure, we come up with the following (Table 7.3) set ofzeroes (FRAs repeated for completeness):

Table 7.3 Synthetic spot zeros and actual forward zeros

Zero rates FRAs(semi) (semi)

6 month 2.50001 year 2.5200 6� 12 2.540018 month 2.5500 12� 18 2.61002 year 2.5750 18� 24 2.6030

As this happens to be the same as in the previous section, we can conclude that the 2-year zero synthesized from coupon rates is too high relative to the 2-year zero synthe-sized from forwards. Our strategy will involve lending at the high 2.575% rate for 2years against borrowing at the 2.5632% rate for 2 years. Let us again assume $100million as the principal of the synthetic 2-year zero. The first part of the strategyremains the same. We:

. Borrow for 18 months by selling $98,715,221.39 face value of an 18-month zeroyielding 2.55% for $95,033,648.12.

. And extend the borrowing for another six months by agreeing to pay 2.603% on$98,715,221.39 of an 18� 24 FRA.

This ensures that we will owe exactly $100 million in 2 years’ time. Against that, wehave to lend synthetically $95,011,591.39 on a zero basis for 2 years in the couponmarket. We:

. Buy $100 million face value of the 2-year 2.5628% par coupon bond and strip thecoupons.

. Sell 2.5628/2¼ $1.2814 face value of a 6-month zero (synthetic or real) yielding2.50%.



. And sell 2.5628/2¼ $1.2814 face value of a 24-month zero (synthetic or real) yielding2.575%.

Each sold zero, if synthetic, would involve additional trades in shorter maturitycoupons.

Exercise Assume that the sold zeros are synthesized from coupons. Refer to Chapter 2to compute face amounts and the cost today. Verify that the total cost, netted againstthe purchase of the $100 million 2-year coupon bond, is $95,011,591.39.


7.5 DYNAMIC HEDGING WITH A EURO STRIP

The arbitrage strategies described in the last two sections when applied to longermaturities can easily involve a large number of securities. A dealer executing manytrades back and forth a day might have to enter into hundreds of largely offsettingsynthetic hedges. This can be cumbersome and result in significant transaction costs. Inthis and the next section, we describe two methods of reducing the dimensionality of thehedge problem. We pretend that all securities, no matter what their maturities orcoupon structures, can be perfectly represented as sets of sensitivities to a fewcommon building blocks. The idea is analogous to our chessboard of contingentclaims of Chapter 1. All securities are viewed as subsets of the entire board, withsome squares overlapping and some not. Furthermore, the board consists of perhapsonly as few as 20 squares. Because of this oversimplification, the replication is not goingto be perfect. The hedge will deteriorate over time and will have to be continuouslyrebalanced. But it will be extremely simple. In this section, the set of canonical hedges isdefined as the set of all ED futures contracts. In the following section, it is a set of a fewselected benchmark bonds. These dynamic hedging techniques are used widely by swap,corporate bond, mortgage bond, and all option traders. The only thing different in eachcase is the set of building blocks.Let us consider the common way of dealing with short-term swaps. The technique is

called blipping the curve (i.e., blipping the ED futures inputs). This starting point is thezero-coupon (discount) curve, obtained by bootstrapping the ED futures. ED futuresare assumed to be the most basic building blocks for all complex instruments; we cansynthesize all other instruments from ED futures. If a resultant synthesized instrumentyields a rate different from the actual one, then that rate can be viewed as ‘‘wrong’’ andarbitrage exists. However, instead of computing the exact amount of each ED contractneeded for the synthetic side of the trade at the outset, the last step involves thecomputation of the sensitivities (durations) of the instrument to a small change ineach ED futures rate, taken one at a time. Each time the ED rate is perturbed, orblipped, the zero curve is rebootstrapped and the value of the synthetic is recomputed.The present value changes relative to the base case (i.e., the sensitivities determine theamount of futures to be entered into as hedges).Suppose we have an opportunity similar to the one described previously, except we

only observe the following (Table 7.4):

Table 7.4 Spot zero rates and ED futures prices

Zero rates ED futures(semi except *)

3 month* 2.454 47 3 month 97.476 month 2.500 0 6 month 97.471 year 2.520 0 9 month 97.47þ18 month 2.550 0 12 month 97.402 year 2.5750 15-month 97.4030 month 2.594 6 18-month 97.413 year 2.621 3 21 month 97.40


where 97.47þmeans 97.475. ED futures allow us to lock in borrowing and lending ratesequal to 100 minus the ED price, quarterly compounded, for 3-month forward periodsstarting on the futures expiry dates. By using the first zero rate with the maturity of 3months and the futures rates starting with the 3-month futures rate (which locks in therate of 100� 97.47¼ 2.53% for the 3� 6 period) and using all the subsequent futuresrates, we can compute, using the familiar recursive formula from Section 6.7; withn ¼ 3, 6, . . . , 24:

�1þ zn

4

�n=3

¼�1þ zn�3

4

�n3� 1�

1þ fn�3�n

4

�

the following set (Table 7.5) of quarterly compounded synthetic zero rates and theirsemi-annual equivalent rates:

Table 7.5 Actual and synthetic zero rates

Zero rates Actual Synthetic Syntheticsemi semi quarterly

3 month 2.454 476 month 2.5000 2.5000 2.492 29 month 2.504 812 month 2.5200 2.5177 2.509 915 month 2.527 918 month 2.5500 2.5480 2.539 921 month 2.547 124 month 2.5750 5.5618 2.553 7

We added the actual semi-annual zero rates in the first column. The biggest discrepancybetween the actuals and the synthetics is in the 24-month maturity.

Let us attempt the usual static arbitrage first. To profit, we borrow for 2 years in thesynthetic market (by borrowing for 3 months and locking in the rates for the rolloverborrowings in the futures market) at 2.5618% and lend in the actual zero market (bybuying a 2-year zero) at 2.575%. We borrow more than we lend for the same face value(today’s discounted value of $1 to be received 2 years from today would be higher usingthe lower synthetic rate than that using the higher actual rate) locking in a sure profit.On the face value of $100 million that profit would be:

100,000; 000=(1þ 0:025 618=2)4 � 100,000,000=(1þ 0:025 750=2)4 ¼ $24,707:60

As before, we work backwards from the $100,000,000 at 2 years to compute theprincipal, or equivalently the number of contracts, for each ED futures by discountingsequentially at the implied forward rate. The results are summarized in Table 7.6:


Table 7.6 Face amounts for forward borrowing

Period Borrow at Face at No. ofstart time end time cars

0� 3 95,036,299 95,619,4583� 6 95,619,458 96,224,251 95.626� 9 96,224,251 96,832,870 96.229� 12 96,832,870 97,444,127 96.8312� 15 97,444,127 98,077,514 97.4415� 18 98,077,514 98,715.018 98.0818� 21 98,715,018 99,354,198 98.7221� 24 99,354,198 100,000,000 99.35

We reborrow the amounts shown in the first column at the start of each period to payoff the loan maturing from the immediately preceding period at the then-prevailing 3-month rate, but we have locked in the net borrowing cost equal to that implied bytoday’s futures prices by shorting futures today. Against this synthetic borrowingstrategy, we buy a 2-year zero yielding 2.575% with a face value of $100,000,000 for$95,011,591.39.Let us now illustrate dynamic arbitrage with the same replicating instruments. The

essence of the curve-blipping method is that instead of statically locking in the profittoday, we commit to a dynamic strategy over the life of the trade. The dynamic strategymeans selling a strip of futures today and rehedging (buying back some or selling more)every day between now and the maturity of the trade (2 years). Dynamic rehedging willgenerate profits and losses over time. The present value (PV) of the difference betweenthe spot trade in the actual security and the sum of all the profits/losses over time isequal to the computed lockable profit. We book a paper profit today, but we realize itover time.Let us take a 2-year zero trade and compute the sensitivity of the value of the 2-year

zero to a 1 basis point (bp) change in the 12-month ED price (¼ rate). Before the blipthe ED rate is 97.40. We assume that the synthetic zero rates are fair. The fair value of$100 million to be received at 2 years is thus $95,036,298.99. Now let us perturb the 12-month ED price to be 97.39 and recompute the set of synthetic zero rates. Thesebecome (Table 7.7):

Table 7.7 Synthetic zero rates with a 1 bpblip to the 12-month ED price

Zero rates Actual Synthetic Syntheticsemi semi quarterly

3 month 2.45456 month 2.5000 2.5000 2.49229 month 2.504812 month 2.5200 2.5177 2.509915 month 2.529918 month 2.5500 2.5496 2.541621 month 2.548524 month 2.5750 2.5631 2.5549


The fair value of $100 million to be received at 2 years changes to $95,033,938.48. Thedollar sensitivity of the 2-year zero price to the 1 bp blip is $2,360.51. Given that eachfutures contract has a sensitivity of $25 per 1 bp, if we short 2,360.51/25¼ 94.42 con-tracts today we will have immunized the value of our 2-year zero to a 1-bp change in the12� 15 forward rate.

We perform the same sequence of calculations for each futures contract, one at atime, arriving at a set of futures contract amounts to be shorted. That is, for eachcontract, we blip the futures price by �0:01, we reconstruct the zero curve, werevalue the 2-year zero with the face value of $100,000,000, take the differencebetween its new blipped value and the original value of $95,036,298.99, and dividethe resultant dollar sensitivity by $25 to get the number of futures to be traded. Thefinal result is given in Table 7.8:

Table 7.8 Hedge of a 2-year zero with an ED futures strip

Futures Recalculated Sensitivity No. ofblipped PV of 2-year cars

zero

3m 95,033,938 2,361 94.446m 95,033,938 2,361 94.449m 95,033,938 2,361 94.4412m 95,033,938 2,361 94.4215m 95,033,938 2,361 94.4218m 95,033,938 2,361 94.4221m 95,033,938 2,361 94.42

Just like in the static lock-in arbitrage strategy, at the outset we would short an entirestrip of ED futures with the amounts shown and sell a 3-month zero (to buy the 2-yearzero). But we would also commit to perform the same blipping procedure every timefutures prices change (for simplicity assume every day) for the next 2 years. Each timewe would compute a new set of contract numbers, compare it with the previous set, andmake adjustments to our positions by trading contracts that need rebalancing. Thelogic is that we are buying the actual 2-year zero unfairly cheaply to yield 2.575%,and dynamic trading will result in cumulating the profit from converging to the fairvalue. The convergence is guaranteed by maturity when the 2-year’s price will be 100.

With the simple cash-and-carry trade like ours, the dynamic hedge will not be verydynamic. We start by shorting futures in the amounts very close to those computed inthe static lock-in. The subsequent adjustments are most likely going to be negligible andour daily profits/losses close to 0. In essence, we are immunizing against fair movementsin the value of the actual 2-year (due to interest rate changes), incurring no profits orlosses, and waiting for the 2-year to adjust its value in the market up to the fair level.Once that happens, our long position will gain in value more than the dynamic hedgewill lose. We can then lift the hedge, sell the zero, and walk away with the profit. (Ofcourse, it can be that the 2-year will not adjust, but the hedge will adjust down, resultingin profit coming from the short side.) Sometimes we have to wait for the full adjustmentall the way to maturity.

The blipping method explained here would rarely be used with a one-off, statically


lockable cash-and-carry arbitrage. However, for large swap or bond portfolios, withnon-coinciding maturity and coupon dates, complex coupon formulae, and mismatchedday-count conventions and coupon frequencies, it is a convenient way of reducing thenumber of hedge instruments to a bare minimum and to ensure that intermediate pointson the yield curve are hedged. As the blipping and hedging is performed with a small setof instruments, the replication requires adjustment over time.The computation of ED sensitivities is also a way of characterizing disparate port-

folios in the same terms. Different portfolios, reduced to the same set of sensitivities,can be compared as to their exposure to different points on the yield curve. This is amore detailed characterization than that obtained by portfolio duration calculations(here we have price values of basis points, or PVBPs, to all quarterly points of the yieldcurve).The discussion in Chapter 9 will make it clear that the blipping method is a version of

the dynamic delta-hedge used by option dealers applied to simpler non-contingentsecurities.

7.6 DYNAMIC DURATION HEDGE

A dynamic duration hedge is a simplified version of the strip hedge, relying on the sameprinciple. We illustrate it by considering a rather obvious example. We get quotes onpar coupon bond yields from a dealer:

5-year 3.00%7-year 5.00%10-year 3.00%

Clearly the 7-year seems to be an obvious candidate for purchasing and the 5-year and10-year for shorting. To profit, first we compute the durations of the bonds. Let usassume that they are the following:

5-year 4.37-year 6.110-year 8.5

To be precise we would want to compute the duration of the 7-year in terms of itssensitivity to the 5-year and the 5� 10 forward and then reconstitute it in terms of theduration with respect to the 5-year and the 10-year. Let us ignore non-parallel curvemovements.We buy $100 million of the 7-year. This position will change in value by 6.1 bp per

1 bp change in yields in the opposite direction. Based on the computed durations weshort $90.5 million of the 5-year and $26 million of the 10-year resulting in zero netsensitivity to yield changes (Table 7.9):


Table 7.9 Duration hedge of a 7-year bond with 5- and 10-year bonds

Bond Duration Face Sensitivity

5-year 4.3 �90.5 �3.897-year 6.1 100 6.1010-year 8.5 �26 �2.21

The next day, as yields change, we recompute the durations and the face amounts sothat we are left with no exposure to yield changes. Every day we realize a net profit orloss on the portfolio which is close to 0 since the portfolio is immunized against yieldcurve movements. We are buying time waiting for a big adjustment in yields. This canbe by the 7-year moving down to yield 3%, realizing a big net gain in our long position,or the 5- and/or 10-years moving up to yield closer to 5%, realizing a big net gain in ourshort positions. If the movement is not big but gradual, we keep holding the position,rebalancing continuously. Over time we accumulate a profit.


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________________________________________________________________________________________________________________________ Swap Markets _______________________________________________________________________________________________________________________

A swap is an agreement to exchange two streams of cash flows: one paid and onereceived. These streams can be customized to suit the parties involved. Usually, theyare designed to resemble coupon interest streams on bonds. As such, the swap is asynthetic exchange of two non-identical securities. The value of the swap is equal to thedifference in the values of the two securities. A swap can also be dissected one cash flowexchange at a time and defined as a set of interest rate or currency forwards packagedtogether to create bond-like coupon exchanges (swaplets). From this perspective,forwards (future cash flow exchanges) are the building blocks for swaps, and thevalue of the swap is equal to the sum of the present values of all the exchanges. Thevaluation of swaps can be made simple or difficult depending on how cleverly we exploitthis dual nature of swaps.To make things easy, we do not immediately jump to definitions, but instead start

this chapter with common applications of swaps in corporate finance decisions. We turnto flow diagrams to abstract from the details of cash flow computation. We begin withthe starkest example of a currency swap, where the two cash flow streams are in twodifferent currencies. We define a plain vanilla interest swap, the most common type ofswap, where the two streams are in the same currency, but one is based on a fixed rateand the other on a floating rate. Then we discuss the pricing and hedging of swaps. Thepricing involves computing the present values of both sides of the swap. The hedging isdone by blipping the underlying discount curves and computing the sensitivities andhedge ratios with respect to a set of predefined hedge instruments. Once we understandthe pricing basics, we look at more complicated applications of swaps, where they arecombined with each other to form complex bonds or they combine asset classes as inequity swaps (synthetic exchanges of bonds for stock baskets). We end with swapmarket statistics. These show the enormous growth of swap markets in the last 20years. By some measures, swaps now represent the largest segment of all financialmarkets.

8.1 SWAP-DRIVEN FINANCE

A corporation that has decided to raise new debt finance faces many choices. It can goto the bank to get a credit line or a revolving loan. The bank (or a syndicate of banks)will set the maximum amount the corporation can borrow and the final maturity of itscommitment. Every month or quarter, the corporation will be able to draw the amountof funds equal to the limit amount or less. If it chooses to draw the funds, it will agree topay an interest rate tied to the short-term borrowing rate, like Prime in the U.S.

domestic credit market or LIBOR in the Eurocurrency market, plus a spread reflectingthe corporation’s credit rating. The corporation will also pay a small fee per month orquarter for the unused amount of funds, simply for the fact that the bank stands readyto provide the funds on demand (commitment fee). From the bank’s perspective, thisarrangement will entail that, any time the corporation draws the loan, the bank willturn to money markets to borrow short term to fund it. If the corporation reduces thelevel of funds used, the bank will repay the borrowed funds. If the corporation keepsborrowing, the bank will roll over its obligations and borrow new funds. The interestrate that the corporation is charged and the bank incurs in the money market floatsmonth to month or quarter to quarter, depending on rollover rates.

Instead of going to the bank, the corporation may decide to issue bonds directly inthe credit markets with the help of an investment banker. Here the corporation faces afew choices. It can issue a fixed coupon bond or a floating coupon bond in the domesticmarket. It can issue a fixed or a floating coupon bond in a foreign market. It can issue abond with a complicated, formula-driven coupon structure or zero-coupon, formula-driven, principal structure. Apart from the corporation’s exact timing and cash flow-matching needs, the structure of the debt will depend on the relative cost of thealternatives.

Suppose a French company, in desperate need of new money, has borrowed heavilyin France in the past and French investors are not demanding new bonds; but Americaninvestors are demanding new bonds and are willing to accept an interest rate 10 bplower than on comparable credits, simply to be able to diversify their risks. Suppose theAmericans want a floating coupon bond because they perceive their interest rates lowby historical standards and want to be able to benefit from potential rises in interestrates. Ideally, the company would like to borrow fixed in France as most of its cashflows are relatively constant and most of its customers pay in euros. Can the companybenefit from the American appetite, but have its liabilities tailored to its needs?

Enter the swap market. The company can sell U.S. dollar-denominated floatingcoupon bonds in the U.S. and enter into a floating dollar–fixed euro currency swapwith a global bank, matching the dates of swap cash flows to the bond coupon dates.

More often than not, the process will work in reverse. Large corporations are on acontinuous hunt to discover cheap sources of finance, whether they need it or not. Therelative cheapness depends on swap rates offered to convert the available source(floating dollar) into a desirable one (fixed euro). The attractiveness of the swap ratesmay make the new issuance desirable, potentially to retire previously issued debt at ahigher cost. This is referred to as swap-driven finance.

Fixed-for-fixed currency swap

A U.K.-based company wants to issue new debt with a maturity of 5 years to finance anexpansion of its American operations. The company is relatively unknown in the U.S.and there is not much demand currently for its debt in the U.S. markets. The companyis expecting that the funds to repay the new debt issue would come from its NorthAmerican operations.

Suppose the current exchange rate is $1.50/£ and the company’s credit rating is AA(i.e., the same as that reflected in swap rates). The company needs to raise $150 millionnow. We observe the following zero-coupon interest rates in the U.S. and U.K. (rates


are usually quoted annual in the U.K. and semi-annual in the U.S., we show bothequivalents):

Table 8.1 Zero rates in the U.K. and U.S.

Term Zero rates Zero rates—————————————— ———————————————U.K. semi U.S. semi U.K. annual U.S. annual

6m 3.000 000 2.200 000 3.022 500 2.212 1001y 3.200 000 2.450 000 3.225 600 2.465 00618m 3.400 000 2.700 000 3.428 900 2.718 2252y 3.600 000 2.950 000 3.632 400 2.971 75630m 3.800 000 3.200 000 3.836 100 3.225 6003y 4.000 000 3.450 000 4.040 000 3.479 75642m 4.200 000 3.700 000 4.244 100 3.734 2254y 4.400 000 3.950 000 4.448 400 3.989 00654m 4.600 000 4.200 000 4.652 900 4.244 1005y 4.800 000 4.450 000 4.857 600 4.499 506

By the arbitrage arguments of Chapter 2 imply that if the company were to issue 5-yearcoupon bonds and wanted to sell them at par, it would have to offer the followingcoupon rates:

Table 8.2 Five-year bond coupon rates

U.K. U.S.

Semi 4.722 118 4.359 746Annual 4.780 222 4.409 975

Suppose that investors do not demand U.S. dollar bonds from the company. If thecompany were to issue them, they would demand rates higher than 4.359 746% semi.The company can obtain dollar financing at that rate by issuing U.K. bonds andswapping their cash flows into dollars. The company issues £100 million face valueof 4.780 222% annual coupon bonds (or 4.722 118% semi). For the 5-year 4.780 222%bond, it receives from investors £100 million; the bonds sell at par. This obligates thecompany to the following set of coupon payments:

Table 8.3 GBP payments

1y 4.780 2222y 4.780 2223y 4.780 2224y 4.780 2225y 104.780 222

The company exchanges the £100 million received from the sale of the bonds into $150million at today’s exchange rate of $1.50/£. At the same time, the company enters into aswap agreement with a financial institution to exchange a stream of GBP-denominated

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cash flows for a stream of USD-denominated cash flows. Specifically, it agrees to thefollowing schedule of cash flows:

Table 8.4 Five-year fixed $-fixed £ swap

Receive in GBP Pay in USD

1y 4.780 222 6.614 9622y 4.780 222 6.614 9623y 4.780 222 6.614 9624y 4.780 222 6.614 9625y 104.780 222 156.614 962

The swap receipts are identical to the coupon and principal payments owed on thebonds and will offset with each other, leaving the company with zero net GBP cashflows. The USD payments are equivalent to 4.409 975% coupon and principal pay-ments on a $150-million face value bond.

The dealer will not charge anything (or pay anything) to enter into this swap with thecompany because the present value of the USD stream is equal to $150 million (basedon the fair par coupon rate in USD) and the present value of the GBP stream is equal to£100 million (based on the fair par coupon rate in GBP), as each is the present value ofa par coupon bond and the two present values are equal to each other based on today’sexchange rate of $1.50/£.

With the bond and the swap in place, the company’s net obligations are identical tothose of a USD $150-million, 5-year, 4.409 975% annual coupon bond. The net of thetransactions is summarized in the following diagram.

Figure 8.1 Summary of the net of transactions.

The receipts from the swap counterparty are passes to the bond investors. What is left isthe dollar obligation in the form of a dollar coupon stream and principal payment atthe end.

Note that the scheduled swap cash flows for year 5 consist on both sides of the swapof the regular coupon interest and the principal repayment. The company’s couponcash flow is £4.780 22 million on the receive side and $6.614 962 (4.409 975% on $150million) on the pay side. The principal exchange is £100 million on the receive side for$150 million on the pay side. In currency swaps, the principal exchange at the end of theswap is standard, unless specifically deleted in the swap agreement. In interest rate


swaps, the principal exchange never takes place as it is in the same currency and is anexact zero-flow offset.Technically, even though swaps can be easily customized to customers’ needs, they

follow certain conventions. In our example, if the company wanted to replicate a dollarbond exactly which would have had semi-annual coupons, as is common with U.S.corporate bonds, it would have most likely issued a GBP bond and swapped it into aGBP–USD, fixed–fixed swap with annual GBP receipts and semi-annual USD pay-ments. For the swap to have been par (i.e., be entered into with no compensatingpayment by either counterparty), the swap’s USD rate would have to be set at4.359 746% semi-annual, instead of 4.409 975% annual. Thus the company wouldhave ended up with annual GBP cash flows on the receive side, which it would havepassed on to its bondholders, and semi-annual USD cash flows on the pay side, its netliability. The cash flows would have been as follows (Table 8.5).

Table 8.5 Five-year fixed $–fixed £ swap


6m 3.269 8091y 4.780 222 3.269 80918m 3.269 8092y 4.780 222 3.269 80930m 3.269 8093y 4.780 222 3.269 80942m 3.269 8094y 4.780 222 3.269 80954m 3.269 8095y 104.780 222 153.269 809

For many counterparties, currency swaps are a way of speculating on currency ratesand interest rate differentials. By swapping into dollars, the company may be gamblingthat the dollar is going to weaken so that the GBP value of its USD liabilities is going todecrease. It may also be fearing that the interest rate differential between the rates in theU.K. and in the U.S. may decrease without an offsetting currency move, so thecompany wants to lock it in today.

Fixed-for-floating interest-rate swap

Let us alter the above example to present a rationale for interest-rate swaps: differentcost of capital in floating and fixed rate markets in the same currency. Our U.K.company now wants to issue new fixed rate debt with a maturity of 5 years tofinance an expansion of its U.K. operations into a government contract business.The funds to repay the new debt issue would come from these new operations. Thecompany knows the schedule of government payments it is to receive. The company haspreviously used bank loans and floating rate debt to match its liabilities to the generalstate of the economy. Investors are comfortable evaluating the company’s floating ratedebt. The company reckons that if it were to issue fixed rate debt it would have to pay a

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premium in the form of a coupon rate above that for corporations with a similar creditrating. The solution is to issue a floating rate bond and swap it into a fixed rate liability.

The company issues a 5-year, £100 million bond whose annually adjusted interestrate is equal to 12-month LIBOR (London interbank offered rate). Recall from priorchapters that such a floating rate bond prices to par as long as each coupon rate is set atthe beginning of the interest accrual period and each interest payment is made at theend of the accrual period. This is because the bond is equivalent to an annuallyrevolving loan where the loan is regranted every year at the then-prevailing 1-yearrate. The company thus receives £100 million (par or face value) for the bond.

The company also enters into a £100-million, notional principal, fixed-for-floatinginterest rate swap whereby it agrees to receive an annual floating coupon stream equalto LIBOR and pay an annual 4.780 222% fixed rate coupon stream. The flows aresummarized in Figure 8.2.

Figure 8.2 Summary of flows.

The company’s LIBOR coupon flows, to the bond holders and from the swap counter-party, are unknown in advance. But they will offset each other exactly. The fixedpayments on the swap and the face value repayment to the bondholders leave thecompany with a net liability equivalent to a fixed rate bond.

Viewing the swap as an exchange of two bonds, all the floating rate coupon receiptson the swap plus £100 million received once in year 5 represent an asset priced to partoday. The 4.780 222% fixed coupon payments on the swap and the £100 million paidonce to the counterparty in year 5 represent a liability priced to par today. No paymentto or receipt from the dealer is needed today to compensate for the difference in thepresent values. The one-time £100-million flows cancel each other and are not made.The remaining flows, floating on the receive side and fixed on the pay side, constitutethe swap.

Fixed and floating rate cash flows do not have to have the same periodicity. Oftentheir structure is determined by market convention. So is the day-count used tocompute the exact flows. In our example, it is more likely that the floating rate bondand the floating interest receipts on the swap would be quarterly, while the fixed couponon the swap would be semi-annual or annual for GBP-denominated bonds and swaps.In the U.S., the fixed coupon would typically be semi-annual while the floating couponwould be set and paid quarterly. In the U.S., the fixed coupon would use a 30/360 day-count while the floating LIBOR side would use Act/360. The quoted semi-annual swaprate would reflect these conventions. Swaps following a market convention for a givencurrency’s interest rates are labeled plain vanilla.

For many counterparties, interest rate swaps are a way of speculating on interestrates. By swapping into fixed, the company may be gambling that the 5-year rates are


going to rise, so it wants to lock the rate in today. (We could also view a decision not toswap as a gamble on the rates going down.)

Off-market swaps

Both prior examples assumed no exchange of any compensatory cash flow upfront.That was due to the fact that the bond issue and the swap occurred at the same time andthe rates on the swap were set to equal current market rates. This need not be the case.Bond issuers can choose to come to the swap market at any time to alter the structure oftheir liabilities back and forth between fixed and floating and one currency vs. another.This may be because of their internal cash flow needs or because they want to speculateon the direction of currency and interest rates.Suppose our U.K. company with American operations issued a 5-year, £100-million,

4.780 222% fixed rate GBP bond a year ago. At that time, the company converted the£100 million proceeds into $150 million to invest in the project, but as it anticipated thewidening of the interest rate differential between the U.K. and U.S. it left the bond issueunswapped. Suppose today the foreign exchange (FX) rate is still $1.5/£, but interestrates in the U.K. have risen by 50 basis points (bp) while interest rates in the U.S. stayedthe same, as in Table 8.6.

Table 8.6 New zero rates in the U.K. and U.S.

Term Zero rates Zero rates—————————————— ———————————————U.K. semi U.S. semi U.K. annual U.S. annual

6m 3.500 000 2.200 000 3.530 625 2.212 1001y 3.700 000 2.450 000 3.734 225 2.465 00618m 3.900 000 2.700 000 3.938 025 2.718 2252y 4.100 000 2.950 000 4.142 025 2.971 75630m 4.300 000 3.200 000 4.346 225 3.225 6003y 4.500 000 3.450 000 4.550 625 3.479 75642m 4.700 000 3.700 000 4.755 225 3.734 2254y 4.900 000 3.950 000 4.960 025 3.989 00654m 5.100 000 4.200 000 5.165 025 4.244 1005y 5.300 000 4.450 000 5.370 225 4.499 506

These rates imply that the 4-year par swap rates in the U.K. and the U.S. are:

Table 8.7 Four-year bond coupon rates

U.K. U.S.

Semi 4.849 306 3.898 814Annual 4.910 531 3.939 253

The company enters into a 4-year GBP–USD fixed–fixed swap. The principal is £100million on the receive side and $150 million on the pay side. The coupon on the receiveside is not the current on-market 4.910 531% rate. Rather, the company asks the

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counterparty to match the bond coupon rate of 4.780 222% which is now off-market(i.e., unfair). As it is willing to accept a rate that is lower than the fair market rate, thecompany will have two choices (the dealer should be indifferent as to which one itchooses). One will be to pay the current on-market USD rate on the pay side of theswap, but receive a one-time payment from the dealer to compensate the company forthe present value differential. The other choice is to receive no payment today, but topay a lower USD rate on the pay side of the swap. Let us examine each choice.

Receive an upfront payment for an off-market swap with a positive PV

The payment will be equal to the present value of the difference between the 4-year parswap rate and the off-market 4.780 222% rate. Using the above zero rates:

Table 8.8 Present value of coupon differences (on- vs. off-market)

Payment difference Present value

1y 0.130 309 0.125 6182y 0.130 309 0.120 1503y 0.130 309 0.114 0244y 0.130 309 0.107 369

Sum of present values 0.467 161

this turns out to be £467,161. This number is also equivalent to £100 million, which isthe value of a 4-year 4.910 531% par coupon bond, minus £99.532 839, which is thevalue of a 4-year 4.780 222% coupon bond:

Table 8.9 Present value of the off-market bond’s cash flows

Payments Present value

1y 4.780 222 4.608 1432y 4.780 222 4.407 5383y 4.780 222 4.182 8104y 104.780 222 86.334 348


The present value of the USD side remains $150 million, which is equivalent to £100million.

Receive nothing upfront and pay a lower USD rate

Instead of being taken out, the £467,161 can be used to lower the present value of theUSD side of the swap by $700,742. That is, the company will pay a USD coupon ratesuch that the present value of those payments and the principal repayment will be equalto $149.299 258 million.


Table 8.10 Present value of the cash flows of a $150 million3.812 193% coupon bond

Payments Present value

1y 5.718 289 5.580 7242y 5.718 289 5.392 9933y 5.718 289 5.160 5974y 155.718 289 133.164 944


The coupon rate is 3.812 193%. Note that the rate is lower than the par rate of3.939 253% by 0.127 060%, which is not equal to the 0.130 309% U.K. rate differential.The company need not stop here. It could take out £467,161 or lock in its USD

coupon to a low rate of 3.812 193%, and then further swap into floating USD byentering into a 4-year USD fixed-for-floating interest rate swap. This additional stepwould leave it with a floating dollar liability for the remainder of the term or until itdecided to swap again. If the company decided to take £467,161 out of the currencyswap, then it would enter into an interest rate swap to receive a par rate of 3.939 253%in USD and to pay LIBOR flat. If it did not take any money out and locked in an off-market 3.812 193% in USD on the fixed–fixed currency swap, then it could enter into aUSD interest rate swap to receive 3.812 193% and pay LIBOR minus a spread whosepresent value in today’s terms is equal to $700,742. The same dealer could offer acombined package of the two swaps in one GBP fixed–USD floating currency swap.

8.2 THE ANATOMY OF SWAPS AS PACKAGES

OF FORWARDS

So far we have viewed swaps as exchanges of two fictitious bonds. In the fixed–fixedcurrency swap, it was an exchange of a fixed rate bond in one currency for a fixed ratebond in another currency. In a plain vanilla interest rate swap, it was an exchange of afixed rate bond for a floating rate bond in the same currency. In a fixed–floatingcurrency swap, it was an exchange of a fixed rate bond in one currency for a floatingrate bond in another currency. If the fictitious bonds’ coupons were set to match marketpar rates, then the exchanges were made with no upfront payments. If one or both ofthe exchanged bonds’ coupons were not set to par rates, then the swap was off-marketwith a potential upfront payment or coupon adjustment to compensate for the devia-tion of the present value of the bonds from par.Instead of treating swaps as exchanges of streams of cash flows, we will now look at

each cash flow exchange. In this light, we will treat swaps as strings of consecutiveforward contracts. For market swaps, each forward may be off-market, but the totalsum of the mark-to-markets (MTMs) (present values) of the component forwards willsum to zero. For all swaps, the present value of the swap will be equal to the sum of theMTMs of the building block forwards. This way of looking at swaps will invokethe same arbitrage relationships as when we built coupon bonds from forward-rate-agreements (FRAs) and FX forwards from deposits.

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Fixed-for-fixed currency swap

Let us examine the currency swap entered into by our U.K.-based company withAmerican operations. The cash flows on the swap are given in Table 8.11:

Table 8.11 Five-year fixed $–fixed £ swap


1y 4.780 222 6.614 9622y 4.780 222 6.614 9623y 4.780 222 6.614 9624y 4.780 222 6.614 9625y 104.780 222 156.614 962

The forward FX rates the company is locking in are equal to the coupon ratios for thecoupon related flows and to the original spot $1.50/£ rate for the final principalexchange:

Table 8.12 Forward FX rates locked in for coupon andprincipal exchanges

Receive in GBP Pay in USD FX forward $/£

1y 4.780 222 6.614 962 1.383 8192y 4.780 222 6.614 962 1.383 8193y 4.780 222 6.614 962 1.383 8194y 4.780 222 6.614 962 1.383 8195y 4.780 222 6.614 962 1.383 8195y 100.000 000 150.000 000 1.500 000

These are not real forwards quoted by anyone. The real forwards that would be quotedcan be arrived at using the spot rate and pairs of zero-coupon rates in the coveredinterest rate parity (CIRP) relationship. Using these actual FX forward quotes, we cancalculate the GBP equivalent of the USD payments that can be locked in today, the netGBP cash flow for each future date that can be locked in today, and thus the MTM oneach forward cash flow exchange locked in the swap. The MTM is the amount thecompany would have to pay upfront if it were to enter into each GBP-for-USD cashflow exchange separately in the FX forward market.

Table 8.13 Swap MTM broken down to individual FX forward MTMs

Receive in Pay in CIRP: FX £ equivalent £ equivalent FX forwardGBP USD forward $/£ of $ outflow received–paid MTM in £

1y 4.780 222 6.614 962 1.488 948 4.442 710 0.337 512 0.326 965 5232y 4.780 222 6.614 962 1.480 936 4.466 743 0.313 479 0.291 888 5973y 4.780 222 6.614 962 1.475 898 4.481 991 0.298 231 0.264 820 8914y 4.780 222 6.614 962 1.473 784 4.488 421 0.291 801 0.245 177 1335y 4.780 222 6.614 962 1.474 561 4.486 054 0.294 168 0.232 057 7545y 100.000 000 150.000 000 1.474 561 101.725 158 �1.725 158 �1.360 909 897

Sum of MTMs 0.000 000


The sum of all the MTMs is equal to 0. The 5-year swap can be viewed as a package ofsix forwards to sell USD forwards for GBP forwards with the amounts set by the swapcash flow schedule as shown in the table. Each forward is off-market (i.e., it has a non-zero PV). The first five have a positive PV to the company. Any dealer willing to takethe other side would have to be compensated by upfront receipts of the MTMs of theforwards. For example, for year 1 the company is to receive £4.780 222 million and pay$6.614 962 million, which at today’s forward FX rate is equivalent to £4.442 710million. Thus the company is scheduled to receive a bargain positive flow of£0.337 512 million. It would have to compensate a dealer by paying him the presentvalue of that amount or £0.326 966 million. The same logic applies to the rest of theflows. The last component forward is a large negative PV contract: the company is toreceive £100 million and pay £101.725 158 million. To accept that, the company wouldhave to be paid £1.360 910 million. Today, the net of all settlements of off-marketforwards is 0.

Fixed-for-floating interest rate swap

Let us turn to the interest rate swap entered into by our U.K. company that issued afloating rate bond to investors. To convert its liabilities to fixed, the company agreed tothe following schedule of cash flows.

Table 8.14 Pay fixed–receive floating GBP swap

Receive in GBP: Pay in GBP12-month LIBOR� 100 million——————————————Set at time Paid at time

1y 0 1 4.780 2222y 1 2 4.780 2223y 2 3 4.780 2224y 3 4 4.780 2225y 4 5 4.780 222

Taking each cash flow exchange at a time, an agreement to receive a floating rate setand paid in such a way in exchange for paying a fixed rate at the end of the interestaccrual period is equivalent to an FRA. Even though we do not know the future cashflows (e.g., the third one will be known in 2 years to be paid in 3 years), we know theequivalent fixed rate that an FRA dealer would let us eliminate that uncertainty into atno cost. That rate is the forward 1-year rate for the interest accrual period. We cancompute the market forward rates for each period as equivalent to floating rate receipts.Then we can proceed in a way analogous to what we did with the currency swap. Wecan compute the differences between the forward-equivalent receipts and the scheduledfixed rate payments, and mark-to-markets on these fixed–fixed exchanges by discount-ing the differences to today. These amounts are how much the company would have topay a dealer to enter into the exchanges of floating-for-fixed (the floating-for-fixedforward costs nothing as that is a fair FRA, but the fixed forward for fixed4.780 222% has a non-zero PV).

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Table 8.15 Swap MTM broken down to individual FRA MTMs

Receive in GBP Pay in GBP Forward rate: Equivalent MTM: PV ofzero PV equivalent received–paid differencesof LIBOR receipts

1y LIBOR set at 0 4.780 222 3.225 600 �1.554 622 �1.506 0432y LIBOR set at 1 4.780 222 4.040 803 �0.739 419 �0.688 4933y LIBOR set at 2 4.780 222 4.860 016 0.079 794 0.070 8554y LIBOR set at 3 4.780 222 5.683 244 0.903 022 0.758 7375y LIBOR set at 4 4.780 222 6.510 494 1.730 272 1.364 944

Sum of MTMs 0.000 000

The sum of the MTMs is 0.Another way to look at this swap is to recognize that each LIBOR receipt balanced

against a fixed forward rate payment has a zero MTM (fair FRA). The remainder of thefixed payment to be made over the forward has value to the swap parties. For example,for the second year the receipt of the floating LIBOR set one year from today to be paid1 year later is covered by £4.040 803 million out of the total of £4.780 222 million paid.The excess of £0.739 419 paid to the swap counterparty 2 years from today can besettled today by an immediate receipt of £0.688 493. The same logic applies to all fiveexchanges. In early years, these have a negative PV to the company; in the last 3 years,they have a positive PV to the company as the forward equivalent of floating receiptsgreatly exceeds scheduled fixed payments.

The interest rate swap can thus be viewed as a package of off-market FRAs settledthrough a set of upfront payments summing up to 0 or a package of on-market FRAsand a string of discount bonds with face values equal to the differences between thestated fixed rate and the forward rates.

Other swaps

It can be stated in general that any swap, currency, interest rate or other can be viewedas a package of forwards with consecutive maturity dates. The forwards in the swap areoff-market, and the sum of the MTMs is equal to the PV of the swap. If the swap is anon-market or a par swap, then the sum of the MTMs on the constituent forwards isequal to zero.

Swap book running

An interest rate swap dealer may trade many swaps a day paying fixed on some andreceiving fixed on others. Similarly, a currency swap dealer may, in any given currency,pay on some swaps and receive on others. Both will deal with hundreds of pay andreceive dates, a variety of quoted rates, on- and off-market, several day-counts, etc.Some pays and receives offset each other partially and some are for similar dates. Thedealers are only left with residual positions to expose them to interest rate or currencyrisks.

At the time of each swap, recognizing that swaps are synthetic packages of forwards,dealers can eliminate their exposure by entering into reverse synthetic transactions in


the forward markets. For example, a fixed rate payer on an interest rate swap may enterinto a string of FRAs on which he agrees to receive a fixed rate. The dealer makes aprofit on the swap by charging a few basis points running (i.e., offering to pay a fewbasis points less on the fixed side of the swap) over the cost of his replicating strategy.This covers the risk that if he is slow to execute the reverse transaction, the zero andforward rates that he uses to compute the swap rate may run away from him and hemay have to transact at less advantageous terms when executing the offsettingsynthetics. The dealer has to transact in offsetting synthetics every time he enters intoa swap. This is very unwieldy and costly.Suppose a dealer has a portfolio of two swaps: a pay-fixed swap maturing on

December 15, 2014, with coupon dates on June 15 and December 15, and a receive-fixed swap maturing on December 3, 2014, with coupon dates June 3 and December 3.If hedged through replication, the dealer would have to enter a lot of nearly offsettingFRAs with close but not identical roll dates.To avoid the execution costs (bid–ask spreads) that static replications would entail

and to reduce the dimensionality of the problem, most dealers choose to run their booksusing dynamic hedging with a small group of most liquid instruments. The dealercomputes his net PV sensitivities resulting from these two swaps or, in general, fromthousands of swaps on his books to a set of common inputs, typically Eurocurrencyfutures and government bonds. The dealer then hedges with that reduced set of instru-ments. This way, all swaps are approximated as constructed from the same buildingblocks. On a daily basis, the PV changes of all the swaps taken together will be exactlyoffset by the sum of the PV changes of the selected set of hedge instruments held inquantities determined by the sensitivities (duration-like hedge ratios).The building blocks are not perfect (only the exact static replication would have

been), and so this approach requires daily rebalancing of the hedge. The rebalancingconsists of computing new sensitivities and adjusting the quantities of the hedge instru-ments held by buying more of some and selling more of some. The advantage ofavoiding thousands of bid–ask spreads on FRAs or futures is enormous. This isreflected in the very tight market-making quotes of dealers in the institutionalmarkets. In a way, they compete with each other by showing the tightest bid–askspread that reflects their lowest costs of manufacturing the swaps.Another great advantage to this approach is in risk definition. The risk of a swap

portfolio can be described with respect to a reduced set of hedge instruments.

8.3 THE PRICING AND HEDGING OF SWAPS

Swap rates always reflect the cost of replicating strategies. We have already shown howthe rate on a currency swap and an interest rate swap is determined using quoted zerorates. In reality, to reflect the actual process of hedging the swaps, the pricing of a swaptypically requires one other initial step: that is the building of the zero curve from therates on the hedge instruments.Here is the thinking process behind the selection of liquid hedge instruments. Zero-

coupons reflecting swap parties’ credits are generally traded only for maturities lessthan 1 year. These are Eurocurrency (LIBOR) deposits that trade freely in both direc-tions (i.e., borrowing and lending at tight spreads). Maturities longer than 1 year have

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to be synthesized from Eurocurrency futures or FRAs. This is achievable with the useof liquid futures or forwards for up to 10 years for the U.S. dollar, 5 years for the euro,and 2 years for the Japanese yen, the Swiss franc, and a few other currencies. Beyondthat, dealers rely on the existence of long-term government bonds that can be boughtand shorted at tight spreads. These bonds reflect the credit rating of the sovereigns, andnot swap parties, and thus require a ‘‘plug’’ of a swap spread which itself is continu-ously determined in the market.

We illustrate the process of pricing a U.S. interest rate swap, its execution, and thesubsequent hedging involved.

Suppose in September of 2003, a New York dealer is contacted by ABC Corp. toprovide a quote on a 5-year, $200-million notional principal, interest rate swap whereABC will pay the dealer a fixed rate, semi-annually, and will receive a floating rate of3-month LIBOR. This is a plain vanilla swap.

Suppose at the time the dealer is faced with the following set of rates on the ‘‘buildingblocks’’. These are taken for this illustration from the September 26, 2003 issue of theWall Street Journal.

Table 8.16 US ‘‘building block’’ rates taken from the Wall Street Journal, Friday,September 26, 2003

Eurodollar (CME) US Treasury issues————————————————————Maturity Coupon Yield

Dec03 98.83 9/30/2005 1.625 1.657Mar04 98.75 9/15/2008 3.125 3.030Jun 98.51 8/15/2013 4.25 4.099Sep 98.19 2/15/1931 5.375 5.000Dec 97.77Mar05 97.35 Money ratesJun 96.96 3-month LIBOR 1.1400%Sep 96.65Dec 96.35Mar06 96.10Jun 95.87Sep 95.65Dec 95.44Mar07 95.25Jun 95.08Sep 94.92Dec 94.77Mar08 94.64Jun 94.51Sep 94.41Dec 94.31Mar09 94.22Jun 94.14Sep 94.07Dec 94.00

We assume that Eurodollar (ED) expiry dates correspond to the swap roll dates. If not,


we would interpolate. First, we need to determine the forward rates that can be lockedin for future 3-month periods. If we had FRAs, we could use those directly. Here we donot. We must correct the forward rates implied in the ED futures prices by what iscalled a futures convexity adjustment. The correction takes into account the fact thatthe MTM settlement on the futures takes place at a constant dollar amount per 1-bpchange in the interest rate every day until the futures expiry, while on a FRA theadjustment is a one-time event on the FRA start date: the longer the expiry date,the bigger the adjustment, and the greater the volatility of the short-term discountrate, the greater the deviation of the MTM present value from that of the forward.1

The ultimate check if the adjustment is correct is the comparison to FRA rates, ifavailable.From the money market and futures (or forward) inputs we build the zero-coupon

curve. That is, we compute the discount factors to a sequence of future dates (3 months,6 months, . . ., 60 months). We also show the set of quarterly compounded zero rates tothose dates.

Table 8.17 Zero-coupon curve derived from ED futures

Forward period time 100ED Convexity FRA rate Zero rate Discount factors(months from today) adjustment (to end time) (to end time)—————————Start End

0 3 1.140 1.140 000 0.997 158 13 6 1.17 0.000 1.170 1.155 000 0.994 249 96 9 1.25 �0.008 1.242 1.183 998 0.991 172 39 12 1.49 �0.016 1.474 1.256 479 0.987 533 312 15 1.81 �0.024 1.786 1.362 327 0.983 143 515 18 2.23 �0.032 2.198 1.501 485 0.977 770 718 21 2.65 �0.040 2.610 1.659 657 0.971 432 121 24 3.04 �0.048 2.992 1.825 959 0.964 219 724 27 3.35 �0.056 3.294 1.988 810 0.956 344 227 30 3.65 �0.064 3.586 2.148 245 0.947 846 830 33 3.90 �0.072 3.828 2.300 660 0.938 861 933 36 4.13 �0.080 4.050 2.446 149 0.929 451 236 39 4.35 �0.088 4.262 2.585 540 0.919 652 339 42 4.56 �0.096 4.464 2.719 426 0.909 502 242 45 4.75 �0.104 4.646 2.847 578 0.899 059 745 48 4.92 �0.112 4.808 2.969 826 0.888 381 348 51 5.08 �0.120 4.960 3.086 624 0.877 500 351 54 5.23 �0.128 5.102 3.198 326 0.866 448 854 57 5.36 �0.136 5.224 3.304 687 0.855 278 857 60 5.49 �0.144 5.346 3.406 508 0.843 998 8

The interpretation of the results is that we know the present value of $1 received orpaid on any future date as simply an outcome of a static strategy to replicate thatreceipt/payment (through lending/borrowing for 3 months and locking in the

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1 The theoretical argument for the convexity adjustment is quite complex and beyond the scope of our discussion.

refinancing rate by buying/selling a strip of ED futures with all intermediatematurities).

We are now ready to price the 5-year swap with a fixed semi-annual rate. ‘‘Pricing theswap’’ here means determining the fixed semi-annual coupon coupon rate that wouldmake a 5-year semi-annual bond have a value of par. The bond would have 10 equalcoupon cash flows, first paid 6 months from today and last 5 years from today and oneprincipal cash flow in 5 years. We want to determine what coupon rate divided by twowould produce the 10 coupon cash flows so that the sum of their present values togetherwith the present value of the principal cash flow at the end would equal to 100. We havealready determined the discount rates and factors for all the semi-annual dates. All weneed to do is to pick a rate and discount cash flows based on it. We can easily verify thatthe rate of 3.3636% is the desired coupon rate. The cash flow discounting is portrayedin the following table.

Table 8.18 PV of 3.3636% coupon bond using derived zero rates

Zero rate Discount factors Par swap rate ¼ 3.3636(to end time) (to end time) ——————————————

Cash flow PV of cash flow

1.140 000 0.997 158 11.155 000 0.994 249 9 1.6818 1.67211.184 664 0.991 167 41.257 978 0.987 518 5 1.6818 1.66081.364 726 0.983 114 21.504 816 0.977 722 1.6818 1.64431.663 938 0.971 359 61.831 202 0.964 119 1 1.6818 1.62151.995 023 0.956 211 22.155 433 0.947 677 4 1.6818 1.59382.308 828 0.938 652 22.455 132 0.929 202 2 1.6818 1.56272.595 214 0.919 3652.729 692 0.909 177 7 1.6818 1.52912.858 358 0.898 698 92.981 055 0.887 985 3 1.6818 1.49343.098 249 0.877 070 23.210 303 0.865 985 6 1.6818 1.45643.316 980 0.854 783 63.419 085 0.843 472 7 101.6818 85.7658

Sum ¼ 100.0000

The dealer quotes ABC Corp. a fixed rate of 3.39%. ABC accepts and the swap is done.ABC Corp. has agreed to pay a fixed rate of 3.39 semi to receive a floating 3-month


LIBOR. The dealer computes his profit relative to his estimated cost of manufacture as2.64 bp running, or 12.28 bp upfront; that is:

0:001 228� 200,000,000 ¼ $245,639

This comes from repricing the swap using the agreed-on rate.

Table 8.19 PV of 3.39% coupon bond using derived zero rates

Zero rate Discount factors Par swap rate ¼ 3.3900(to end time) (to end time) ——————————————

Cash flow PV of cash flow

1.240 000 0.997 158 11.155 000 0.994 249 9 1.6950 1.68531.184 664 0.991 167 41.257 978 0.987 518 5 1.6950 1.67381.364 726 0.983 114 21.504 816 0.977 722 1.6950 1.65721.663 938 0.971 359 61.831 202 0.964 119 1 1.6950 1.63421.995 023 0.956 211 22.155 433 0.947 677 4 1.6950 1.60632.308 828 0.938 652 22.455 132 0.929 202 2 1.6950 1.57502.595 214 0.919 3652.729 692 0.909 177 7 1.6950 1.54112.858 358 0.898 698 92.981 055 0.887 985 3 1.6950 1.50513.098 249 0.877 070 23.210 303 0.865 985 6 1.6950 1.46783.316 980 0.854 783 63.419 085 0.843 472 7 101.6950 85.7770

Sum ¼ 100.1228Profit ¼ 0.1228

In order for the dealer to realize this profit, he will need to manufacture the swap at theestimated cost. That is, he will have to hedge the PV change on the swap so that everytime rates change the PV change on the swap will be exactly offset by the change in thePVs of the hedge instruments held. Immediately after or at the time of the swap, thedealer has to sell a strip of futures. The amounts are computed by blipping the curveone instrument at a time. Let us change the price of the first ED Dec03 contract by�0:01 to 98.82. We rebuild the zero curve and reprice the swap. The change in thepresent value of the swap in dollars divided by $25 gives us the number of Decembercontracts that need to be sold to hedge the swap.

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Table 8.20 PV of 3.39% coupon bond using zero rates derived by blipping Dec03 ED contract

Forward period time 100ED Convexity FRA rate Zero rate Discount factors Par swap rate ¼ 3.3900 0.026(months from today) adjustment (to end time) (to end time) ——————————————————— Cash flow PV ofStart End cash flow

0 3 1.140 1.140 000 0.997 158 13 6 1.18 0.000 1.180 1.160 000 0.994 225 1 1.6950 1.68526 9 1.25 �0.006 1.244 1.187 998 0.991 142 79 12 1.49 �0.012 1.478 1.260 479 0.987 493 9 1.6950 1.673812 15 1.81 �0.018 1.792 1.366 727 0.983 089 615 18 2.23 �0.024 2.206 1.506 484 0.977 697 6 1.6950 1.657218 21 2.65 �0.030 2.620 1.665 369 0.971 335 421 24 3.04 �0.036 3.004 1.832 454 0.964 095 1.6950 1.634124 27 3.35 �0.042 3.308 1.996 137 0.956 187 427 30 3.65 �0.048 3.602 2.156 435 0.947 653 8 1.6950 1.606330 33 3.90 �0.054 3.846 2.309 739 0.938 628 833 36 4.13 �0.062 4.068 2.455 968 0.929 179 1 1.6950 1.575036 39 4.35 �0.070 4.280 2.595 986 0.919 342 139 42 4.56 �0.078 4.482 2.730 409 0.909 155 1.6950 1.541042 45 4.75 �0.086 4.664 2.859 027 0.898 676 545 48 4.92 �0.094 4.826 2.981 683 0.887 963 2 1.6950 1.505148 51 5.08 �0.102 4.978 3.098 840 0.877 048 351 54 5.23 �0.110 5.120 3.210 862 0.865 964 1.6950 1.467854 57 5.36 �0.118 5.242 3.317 510 0.854 762 357 60 5.49 �0.126 5.364 3.419 588 0.843 451 6 101.6950 85.7748

OriginalSum ¼ 100.1203 100.1228

PV change in % of par (0)PV change in $ (4,991)

No. of cars (200)

Next, we return the Dec03 futures value back to 98.83 and change the price of theMar04 contract by �0:01 to 98.74. We rebuild the curve and reprice the swap. Thechange in the present value of the swap in dollars divided by $25 gives us the number ofMar04 contracts that need to be sold to hedge the swap.

Table 8.21 PV of 3.39% coupon bond using zero rates derived by blipping Mar04 ED contract

Forward period time 100ED Convexity FRA rate Zero rate Discount factors Par swap rate ¼ 3.3900 0.026(months from today) adjustment (to end time) (to end time) ——————————————————— Cash flow PV ofStart End cash flow

0 3 1.140 1.140 000 0.997 158 13 6 1.17 0.000 1.170 1.155 000 0.994 249 9 1.6950 1.68536 9 1.26 �0.006 1.254 1.187 997 0.991 142 79 12 1.49 �0.012 1.478 1.260 478 0.987 493 9 1.6950 1.673812 15 1.81 �0.018 1.792 1.366 726 0.983 089 715 18 2.23 �0.024 2.206 1.506 484 0.977 697 7 1.6950 1.657218 21 2.65 �0.030 2.620 1.665 369 0.971 335 421 24 3.04 �0.036 3.004 1.832 454 0.964 095 1 1.6950 1.634124 27 3.35 �0.042 3.308 1.996 137 0.956 187 427 30 3.65 �0.048 3.602 2.156 435 0.947 653 8 1.6950 1.606330 33 3.90 �0.054 3.846 2.309 739 0.938 628 833 36 4.13 �0.062 4.068 2.455 968 0.929 179 1 1.6950 1.575036 39 4.35 �0.070 4.280 2.595 986 0.919 342 139 42 4.56 �0.078 4.482 2.730 409 0.909 155 1.6950 1.541042 45 4.75 �0.086 4.664 2.859 027 0.898 676 545 48 4.92 �0.094 4.826 2.981 682 0.887 963 2 1.6950 1.505148 51 5.08 �0.102 4.978 3.098 840 0.877 048 351 54 5.23 �0.110 5.120 3.210 862 0.865 964 1.6950 1.467854 57 5.36 �0.118 5.242 3.317 509 0.854 762 357 60 5.49 �0.126 5.364 3.419 588 0.843 451 7 101.6950 85.7748

OriginalSum ¼ 100.1204 100.1228

PV change in % of par (0)PV change in $ (4,906)

No. of cars (196)

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We return the Mar04 futures value back to 98.75 and change the price of the Jun04contract by �0:01. We continue the process, each time rebuilding the curve and re-pricing the swap. We observe the changes in the present value of the swap. We dividethese by $25 to get the number of all contracts that need to be sold to hedge the swap.Starting with Sep08 the PV changes will be 0. The results are summarized in Table 8.22.

Table 8.22 Summary of results (EDs on the CME)

Hedge No. of cars Hedge No. of cars Hedge No. of cars

Dec03 �200 Dec �185 Dec �172Mar04 �196 Mar06 �182 Mar08 �169Jun �196 Jun �182 Jun �169Sep �193 Sep �179 Sep 0Dec �192 Dec �179 Dec 0Mar05 �189 Mar07 �175 Mar09 0Jun �189 Jun �175 Jun 0Sep �185 Sep �172 Sep 0

Dec 0

We also compute the hedge for the change in the spot LIBOR. A 1bp blip in the 3-month spot rate results in a swap PV change of $�4,992, which is equivalent to �200contracts. Trading in spot deposits can be costly, so dealers normally hedge this partwith the first available ED contract as a proxy for spot (a serial October contract ifavailable, or stack on top of the other December contracts) or leave it unhedged.

Table 8.23 Additional hedge(ED on the CME)

Hedge No. of cars

Oct03 �200

Each day after the inception of the swap and every time interest rates change dramatic-ally, the dealer will repeat the entire exercise to compute the new hedge amounts andtrade the appropriate number of contracts to ensure that his holdings are equal to thecurrently computed hedge. Every day the profits and losses on the futures will exactlyoffset the PV changes on the swap. As the swap ages and actual cash flows drop off,fewer maturities of the futures will be held. Daily adjustments to the positions will besmall, but over time all hedges will be liquidated.

8.4 SWAP SPREAD RISK

Swap spread is defined as the difference between the par rate on a swap of a givenmaturity and the yield on a risk-free government security with the same maturity. Inour example above, the fair 5-year swap rate was 3.3636% while the 5-year yield on the5-year U.S. Treasury was 3.03. The 5-year swap spread in this case was 33.4 bp. Dealerscontinuously quote the spreads over governments they are willing to pay (bid) andreceive (ask) on swaps with all standard maturities. The spread is quoted on thesame basis as the governments (e.g., semi-annual 30/360 in the U.S., semi-annualAct/365 in the U.K, annual Act/Act for German Bunds and French OATs). The

assumption is that the underlying government rates are quite variable while the spreadsare not, so it is convenient to quote just the ‘‘add-ons’’. But at the time a swap isexecuted, the actual swap rate is agreed on. In the U.S., all annual maturities up to15 years plus the 30-year are shown routinely on financial screens. All other maturitiesare quoted on request. In the U.K., maturities up to 5 years and the 10-year are quotedroutinely. In Japan, maturities up to 2 years and the 10-year are quoted routinely. Theavailability of the hedge instruments dictates which maturities are considered liquidenough to make markets in.

Swap spread risk refers to the possibility that the PV of the swap will not be offset bythe PV change in the underlying hedge instruments because those instruments aregovernment securities with no default premium allowed for in the yield. Let us illustratethis by extending the above example.

Suppose the dealer had quoted the 5-year spread of 33/36. ABC Corp. accepted theoffer and agreed to pay 36 bp over the current 5-year Treasury note yielding 3.03% (i.e.,3.39%). The dealer could have chosen to hedge the 5-year receive-fixed swap, by short-ing the 5-year government note. He would have had to compute the change in the valueof the swap and the change in the price of the 5-year note resulting from a 1-bp changein the yield to maturity. The ratio of these two sensitivities would have given him theamount of the 5-year note to be shorted. The procedure would be repeated every dayensuring that swap PV changes would be offset by Treasury PV changes. This wouldonly be true if yields on swaps moved one for one with yields on Treasuries (i.e., only ifthere were no swap spread changes). If the 5-year swap spread were to rise, resulting inthe swap yield moving up by more than the Treasury yield or the swap yield movingdown by less than the Treasury yield the dealer would incur a loss.

The ED hedge chosen by the dealer for the 5-year swap did not expose him to thespread risk, because ED contracts reflect the same credit as swaps (i.e., they implicitlyinclude a default premium).

Most dealers get naturally exposed to swap spread risk on swaps with long matur-ities. Eurocurrency contracts are neither available nor liquid enough, and dealers haveno choice but to use government securities as hedges. Governments are also moreconvenient as they require one trade instead of many, potentially saving cost andeffort. For that reason and for speculative reasons, dealers choose them for swapswith shorter maturities, too.

A dealer computes and monitors closely his exposure to swap spreads of all matu-rities. Often he actively seeks out swaps that might reduce those exposures by offeringtighter spreads for the desired maturity. Swap spread risk is the secondary risk swap adealer is left with after hedging the primary interest rate risk. It can be his strategicadvantage or disadvantage when competing with other dealers. It is the main risk of awell-run swap book.

8.5 STRUCTURED FINANCE

Just as forwards and zeros can be viewed as building blocks for swaps, swaps can beviewed as building blocks for structured bonds. We showed in the swap-driven financeexamples how one liability can be turned into another with the use of swaps: fixed intofloating, one currency into another, etc. In those situations, corporations use swaps atthe time of or subsequent to a bond issuance to match the demand of their bond


investors to their own funding needs. This wizardry can take advanced forms bycombining several swaps with each other and with options to create highly complicatedbond structures (we discuss option pricing later). The driver here is investor demand fora particular exposure scenario. The issuer’s objective is simply a lower cost of financing.

Inverse floater

As the first example we consider an inverse floater. This is a bond whose coupon payoutis equal to a formula of some fixed rate minus a short-term floating rate. Such couponstructure is often demanded during times of low rate volatility, a steep yield curve, andwhen investors expect the short-term rates to decrease. The pricing is attractive then, asforwards increase with maturity.Using the rate inputs as of September 26, 2003 an investor may be offered by ABC

Corp. a 5-year bond whose quarterly coupon is equal to:

6:52� LIBOR3m

Let us explain this structure in a diagram.

Figure 8.3 An inverse floating bond.

On the issue date ABC sells the inverse floater bonds with a coupon of 6:52� L3m for100. It enters into a swap with a notional principal of 200 to receive 3.36 and pay L3m

on the same dates (times the appropriate day-count). At the same, it buys a cap forwhich it pays upfront or over time the equivalent of 8 bp running. The cap providesprotection for the scenario that L3m is above 6.52 on any coupon date. In those cases,the option dealer who provided the cap will compensate ABC for the difference. If L3m

stays below 6.52, ABC will receive nothing. ABC nets 12 bp running which reduces itscost of funding. The swap dealer makes a profit on the swap (paying 3.36 while the fairrate is 3.3636); the option dealer makes a profit on the cap. The investor gets a desiredcoupon formula, paying well above today’s 1.14 LIBOR. To make the cash flows clear,let us examine two scenarios.First, assume that on some coupon date L3m is 1.25 (i.e., lower than 6.52). The bond

holder coupon is 5.27. ABC receives 3.36 on the two swaps (i.e., a total of 6.72). It pays

Swap Markets 219

0.08 out of that for the cap and keeps 0.12. With the remainder, equal to 6.52, it pays1.25 twice on the two swaps and 5.27 to the bond holders (i.e., it pays exactlyL3m ¼ 1:25 minus the 0.12 it kept).

Next, assume that L3m is 8.50 (i.e., higher than 6.52) on some coupon date. The bondholder coupon is 0. ABC receives 3.36 on the two swaps (i.e., a total of 6.72). It pays0.08 out of that for the cap and keeps 0.12. The remainder is equal to 6.52, but ABCalso collects 1.98 on the cap. It pays the sum of the two, equal to 8.50, on the first swapand another 8.50 out of its own pocket on the other swap.

The net financing cost to ABC is L3m minus 0.12 irrespective of the level of L3m onany coupon date. The coupon on the bond looks attractive to most investors who donot believe in short-term rates going up from the currently low level. The fixed portionof the coupon reflects double the current term swap rate net of the cost of the cap. Capprotection tends to be cheap as it has a high strike price. In most cases, the swap and theoption dealer are the same, providing the issuer with a competitive quote on the entirebehind-the-scenes part of the deal (in our case the 3.36 swap level is very close to the fairlevel and the dealer makes most of his profit on the cap).

Leveraged inverse floater

The leveraged inverse floater combines three or more swaps and a cap into a singlebond. Consider the following quarterly coupon formula offered by ABC to its investors:

9:80� 2� L3m

It is easy to see that, compared with the leveraged floater, all we need to do is to addanother swap and lower the strike on the cap. Here is the summary diagram.

Figure 8.4 A leveraged inverse floating bond.

The structure works for everyone involved. The investors’ coupon on day 1 is9:80� 2� 1:14 ¼ 7:52, which is very attractive compared with the alternatives of3.36 fixed or L3m floating. The dealer sells a cap and enters into a profitable swapwith a notional principal equal to three times that on the bond. The issuer gets


L3m � 10 bp financing. The investors bear the risk in this structure that short ratesincrease rapidly, which would sharply reduce or potentially eliminate the couponpayout.2

Capped floater

A capped floater is a bond whose coupon is floating but capped. The investor gives uppotentially increased coupon payouts for a larger coupon now. The issuer desiresreduced cost of financing. This structure is simpler than an inverse floater.Suppose investors demand a floating rate. For a little extra spread on top of the

floating rate, they are willing to give up upside potential. If ABC needs floating ratefinancing, then it can simply sell the cap embedded into the coupon formula to thedealer and use the proceeds to enhance the coupon offered. If ABC needs fixed ratefinancing, then it can purchase a cap and swap the floating rate on the bond into a fixedliability. The next diagram shows this more complicated scenario.

Figure 8.5 A capped floating bond.

The investor’s coupon is L3m plus 8 bp, but no more than 6.00. ABC get financing equalto 3.35 fixed. The swap dealer receives the ask side of the fixed rate, 3.39, and paysLIBOR. The option dealer buys a cap. If LIBOR on any coupon date is below 5.92, thecap does not pay. ABC simply passes the L3m coupon it receives on the swap to thebond holders, enhanced by 8 bp out of the 12 bp it receives as the payment for the cap.If LIBOR on a coupon date is greater than 5.92, say 7.00, the situation is only a littlemore complicated. ABC receives 7.00 from the swap dealer and passes 6.00 to the bondholders. It also receives 0.12 from the cap dealer. Out of the 1.12 total, it pays7:00� 5:92 ¼ 1:08 on the cap back to the option dealer, leaving it with 0.04 whichreduces its net fixed rate liability.

Callable

Fixed rate bonds, floating rate bonds, and all the structured bonds can be issued in acallable form. In this case, ABC has the right to call the bonds from the bond holders atpar or a price specified in advance in a call schedule. ABC can choose to retain that

Swap Markets 221

2 Consider the duration and convexity of a leveraged inverse floater. A small rate increase may render the PV of futurecoupons 0.

right (in which case it will bear the cost of a higher coupon demanded by the investors)or sell it to the dealer (in which case the dealer will cover that cost, leaving ABC with astraight fixed rate liability). The choice depends on ABC’s needs and on the deal’spricing. If interest rates are currently volatile, the dealer will be willing to pay a lotfor the call right, potentially reducing ABC’s fixed financing cost in a significant way.

Range

A range bond pays a coupon that depends on how many days during the interestaccrual period a floating rate stayed within a pre-specified range. In this case, theissuer simply desires low cost of floating financing. It offers investors an enhancedfloating (LIBOR plus spread) or fixed coupon rate for each day the floating ratedoes not leave the range bounds. Investors get no interest rate accrual for days whenthe rate goes outside the range. They are betting that the rate will not jump up and/ordown. The range can have upper or lower bounds or both. The coupon formula mightread as follows:

L3m þ 100 bp but only for days where 0:75 < L3m < 2:75

The range option embedded in the coupon is sold to an option dealer in exchange forthe spread and some extra that reduces the issuer’s financing. This is similar to a cappedfloater except the option is a lot more complicated from the dealer’s perspective.

Index principal swap

A structure attractive to mortgage bond investors who want the enhanced yield of aprepayment-exposed investment, but want to limit the unpredictability of prepaymentsis called an index principal swap (IPS) or index-amortizing swap (IAR). As is often thecase, this structure is driven by this specific investor demand, while the issuer simplywants reduced floating financing. The issuer issues a bond with a formula coupon andprincipal to the bond holders, but reverses the cash flows of the bond by entering intoan IPS with a dealer.

The swap is a fixed-for-floating swap, except the notional principal of the swapchanges over time. To compute the principal for each period, one takes the principalas of the last period times one minus a percentage taken from an amortization table thatmight look like this:

Rate Percent of notionalamortized

3 02 501 100

Suppose the last period’s principal was 100 and the index rate L3m of this period is1.14%. We use 1.14 to interpolate between 1 and 2 in the table to obtain the notionalpercentage equal to 93%. This period, the new notional principal used to computeinterest flows on both sides of the swap is equal to the last principal times 100 minusthe amortized percentage of 93, or 0.07. The new principal is thus 7.


This procedure is intended to mimic the prepayment behavior of fixed rate mort-gages. As rates go down, homeowners refinance their mortgages. Mortgage bonds’principals are reduced. Similarly, IPS bonds principals will be reduced, but based ona pre-specified amortization table rather than the actual behavior of homeowners whooften do not refinance optimally.The IPS was first popularized in the mid-1990s in the U.S. where over 50% of

morgages are fixed, with no prepayment penalties, and a large securitized mortgagebond market exists.

8.6 EQUITY SWAPS

Some investors desire equity market exposure in a bond form. This can be accom-plished by defining the formula for a bond coupon payout as the appreciation of thestock index over the accrual period times the principal of the bond and times theappropriate day-count fraction. In this case, the issuer of the bond desires a low costof financing that has nothing to do with equities. The dealer packages the bond byentering into an equity swap with the issuer. In the equity swap, every coupon period,the dealer pays the stock index appreciation on a notional principal of the swap andreceives a floating interest rate (e.g., LIBOR) times the appropriate day-count fractiontimes the same notional principal. The stock return each period is passed on to the bondholders in the form of a bond coupon. The LIBOR-based interest is the issuer’s net costof financing. The structure naturally includes an at-the-money option given that thecoupon paid to the bond holder cannot be negative. Alternatively, the bond holder getsa fixed rate plus only a predefined portion of the appreciation combined with out-of-the-money options. The option is paid for by the bond holder through a reducedcoupon formula (negative spread).Here is a diagram for an S&P 500 index-based coupon assuming the (European) put

cost is 1.3% running. SPX stands for the percentage change in the S&P 500 for thecoupon year. The investor receives the excess of SPX over 2%, floored at zero.

Figure 8.6 An S&P 500-linked coupon bond.

Let us distinguish two scenarios. First, let us look at SPX> 2% (e.g., SPX ¼ 14%). Theinvestor gets a 12% coupon. ABC receives 14% on the swap, pays 1.8% for the put,receives no payout from the put, and keeps 0.20%, which reduces its floating financingcost. Second, let us look at SPX< 2% (e.g., SPX ¼ 0.5%). The investor gets a 0%

Swap Markets 223

coupon. ABC receives 0.5% on the swap and 1.5% payout from the put, pays 1.8% forthe put, and keeps 0.20%, which reduces its floating financing cost. Similarly, if SPXturns out negative for the year, the bond holder gets no coupon and ABC keeps a0.20% reduction in its financing cost.

The pricing on the swap works as long as we can show, similarly to what we haveestablished for the fixed-for-floating par swap, that the SPX-for-L3m swap is fair (i.e., ithas a zero PV).

This follows directly from the cash-and-carry relationship for stock index futures.The equity swap can be viewed as a string of 1-year synthetic forward (or futures)contracts. In our case, from the dealer’s perspective, the swap’s cash flows—paySPX, receive LIBOR—are equivalent to a string of short index forwards or reversecash-and-carry transactions. In the reverse cash-and-carry transaction, the dealerborrows a basket of stocks, sells the stocks short, and invests the proceeds to earnLIBOR. At the end of the year, he covers the short by buying the basket in the spotmarket. The initial sale and subsequent repurchase of stocks are equivalent to payingout the appreciation on the basket. The interest on short-sale proceeds is equivalent toreceiving floating LIBOR. In order to hedge the string of short forwards packaged intothe swap, the dealer has to hedge by doing the opposite (i.e., he has to receive the SPXappreciation and pay LIBOR). He does that by entering each year into a 1-year cash-and-carry. He borrows the principal to pay LIBOR. He buys the stocks at the beginningof each year and liquidates them on each coupon payout date. The profits on stocks arepassed to ABC to satisfy the swap obligation. The LIBOR receipt from ABC covers theborrowing cost. Each year the dealer repeats the exercise for the following year. (Inreality, on the coupon pay date, the dealer liquidates only the appreciated part of theholdings and pays the proceeds to ABC and holds the rest for the following year.)

8.7 COMMODITY AND OTHER SWAPS

There are parallels of equity swaps that involve other asset classes. They are alwaysfixed income hybrids in that one side of the swap is a floating interest rate stream, whilethe other is some other asset return stream. A common example is a commodity swap,potentially linked to a bond issuance, whose coupon payout is a formula based on apercentage change in the price of a commodity over the coupon period. Structures soldin the past 10 years have involved oil, gold, and a variety of financial assets (e.g.,weather-related contracts, total-rate-of-return contracts). Many have been partlyhedgeable with existing exchange-traded or over-the-counter (OTC) forward andfutures structures. All have been statically replicable with cash-and-carry purchasesof the underlying assets against borrowing in the LIBOR markets.

Increasingly, dealers push the envelope of innovation by looking to secondary andunhedgeable variables on which to write swaps. One such variable is the historicalvolatility of an interest rate or a price; another is the inflation rate in an economy.There is no market in volatility or inflation forwards; one side of the swap cannot besynthetically replicated or dynamically hedged. The trading of such products is moreakin to bookmaking and insurance sales than to swap trading. In most cases, riskmanagement relies on book-matching or diversification. If both sides of the trade can


be found and the dealer can find two counterparties willing to take the opposite sides ofswap, then the dealer simply extracts the bid–ask spread on the bet the way a book-maker does on a sporting event. If the offsets are not identical, then the dealer tries todiversify the risks across the events and calendar dates, in effect pooling unhedgeablerisks the way an insurer does.

8.8 SWAP MARKET STATISTICS

The global OTC derivatives market is enormous. According to a paper in the BISQuarterly Review of June 2003 titled ‘‘International banking and financial marketdevelopments’’, the total notional principal outstanding reached over $140 trillion bythe end of 2002.The biggest category is interest rate derivatives accounting for over $100 trillion. Of

that, euro-denominated interest rate swaps accounted for over $30 trillion and U.S.dollar-denominated swaps for under $25 trillion. The vast majority of swaps are under5 years of maturity, with only 20% over that mark. That, however, includes aged long-term swaps. The maturity differential is also related to the availability of hedge instru-ments in various currencies.

Table 8.24 Notional amounts outstanding of OTC derivatives by risk category and instrument(in billions of U.S. dollars)

Dec. 1998 Dec. 1999 Dec. 2000 Dec. 2001 Dec. 2002

Foreign exchange 18,011 14,344 15,666 16,748 18,469Outright forwards and FX swaps 12,063 9,593 10,134 10,336 10,723Currency swaps 2,253 2,444 3,194 3,942 4,509Options 3,695 2,307 2,338 2,470 3,238

Interest rate contracts 50,015 60,091 64,668 77,568 101,699Forward-rate agreements 5,756 6,775 6,423 7,737 8,792Interest-rate swaps 36,262 43,936 48,768 58,897 79,161Options 7,997 9,380 9,476 10,933 13,746

Equity-linked contracts 1,488 1,809 1,891 1,881 2,309Forwards and swaps 146 283 335 320 364Options 1,342 1,527 1,555 1,561 1,944

Commodity contracts 415 548 662 598 923Gold 182 243 218 231 315Other commodities 233 305 445 367 608Forwards and swaps 137 163 248 217 402Options 97 143 196 150 206

Other 10,389 11,408 12,313 14,384 18,337

Total contracts 80,318 88,202 95,199 111,178 141,737

Source: http://www.bis.org/publ/qcsv0306/anx1920a.csv

Most single-currency interest rate swaps and FX derivatives were dealer-to-dealer ordealer-to-other financial institution.

Swap Markets 225

Table 8.25 Notional amounts outstanding of OTC single-currency interest rate derivatives byinstrument and counterparty (in billions of U.S. dollars)


Forward rate agreements 5,756 6,775 6,423 7,737 8,792With reporting dealers 2,848 3,790 3,035 3,658 4,579With other financial institutions 2,384 2,596 2,851 2,955 3,540With non-financial costumers 523 389 537 1,124 673

Swaps 36,262 43,936 48,768 58,897 79,161With reporting dealers 18,310 23,224 24,447 27,156 36,321With other financial institutions 13,971 16,849 20,131 25,197 34,383With non-financial costumers 3,980 3,863 4,190 6,545 8,457

Options 7,997 9,380 9,476 10,933 13,746With reporting dealers 3,283 3,503 4,012 4,657 5,781With other financial institutions 3,435 4,566 4,066 4,358 5,684With non-financial costumers 1,279 1,310 1,399 1,918 2,281

Total contracts 50,015 60,091 64,668 77,568 101,699With reporting dealers 24,442 30,518 31,494 35,472 46,681With other financial institutions 19,790 24,012 27,048 32,510 43,607With non-financial costumers 5,783 5,562 6,126 9,586 11,411

Source: http://www.bis.org/publ/qcsv0306/anx21a21b.csv

Table 8.26 Notional amounts outstanding of OTC FX derivatives by instrument and counter-party (in billions of U.S. dollars)


Outright forwards and FX swaps 12,063 9,593 10,134 10,336 1,072With reporting dealers 5,203 3,870 4,011 3,801 431With other financial institutions 5,084 4,123 4,275 4,240 436With non-financial costumers 1,777 1,600 1,848 2,295 204

Currency swaps 2,253 2,444 3,194 3,942 450With reporting dealers 565 651 881 1,211 141With other financial institutions 1,024 1,072 1,410 1,674 190With non-financial costumers 664 721 904 1,058 118

Options 3,695 2,307 2,338 2,470 323With reporting dealers 1,516 871 838 900 110With other financial institutions 1,332 908 913 842 133With non-financial costumers 847 529 588 728 80

Total contracts 18,011 14,344 15,666 16,748 1,846With reporting dealers 7,284 5,392 5,729 5,912 683With other financial institutions 7,440 6,102 6,597 6,755 760With non-financial costumers 3,288 2,850 3,340 4,081 403

Source: http://www.bis.org/publ/qcsv0306/anx1920a.csv

Most single-currency interest rate swaps and FX derivatives were dealer-to-dealer ordealer-to-other financial institution.


Figure 8.8 Interest rate swaps, notional amounts outstanding (in trillions of U.S. dollars): (a) bycurrency; (b) by counterparty; and (c) by maturity. The latter includes FRAs, which in December,2002 accounted for approximately 6% of the total notional amount outstanding.From BIS Quarterly Review, June, 2003, ‘‘International banking and financial market developments’’.

While dealer-to-dealer trades offset some risks, exchange-traded short-term interestrate futures, and in particular Eurocurrency (Eurodollar, Euroeuro, Euroyen, etc.)contracts, remain the main hedge instrument of choice for the residual exposure onmost swaps under 5 years (under 10 years in the U.S.). This is evidenced in thecontinuing growth in the turnover of those contracts among other futures andoptions, representing around $100 trillion worth of contracts per quarter. These aresplit equally between the dollar and the euro-denominated interest rate contracts.

Swap Markets 227

Figure 8.7 Notional amounts outstanding by broad risk category ($ trillions) and gross marketvalues by broad risk category ($ trillions). { Estimated positions of non-regular reporting institu-tions.Source: BIS

Figure 8.9 Turnover of exchange-traded futures and options, quarterly date (in trillions of U.S.dollars): (a) by contract type and (b) by region.Sources: FOW TRADE data, Futures Industry Association, BIS calculations. From BIS Quarterly Review, June, 2003,

‘‘International banking and financial market developments’’. Reprinted with permission from BIS.

For long-term swaps, spot long-term government bonds are the main hedge instru-ments. While the turnover of the note and bond futures has increased in the U.S.,this has not been the case universally. Swap hedgers have always preferred spotbonds to futures in order to avoid delivery and embedded option risk.

Figure 8.10 Turnover in government bond contracts, quarterly futures contract turnover (intrillions of U.S. dollars): (a) U.S.; (b) Germany; and (c) Japan and the U.K.Sources: FOW TRADE data, Futures Industry Association, BIS calculations. From BIS Quarterly Review, June, 2003,

‘‘International banking and financial market developments’’. Reprinted with permission from BIS.

The post-technology boom era of the early 2000s has ushered in very low short-terminterest rates, resulting in fairly steep swap yield curves for most currencies, even inJapan where interest rates on all maturities have remained under 1%.


Swap Markets 229

Figure 8.11 Swap yield curves (in percentages): (a) U.S. dollar; (b) euro; and (c) yen.Note: For 3-, 6-, and 12-month U.S. dollar and yen maturities, LIBOR; for 3- and 6-month euro maturities, euro deposit

rates. Source: Bloomberg. From BIS Quarterly Review, June, 2003, ‘‘International banking and financial market develop-

ments’’. Reprinted with permission from BIS.

______________________________________________________________________________________________________________________________________ Part Three ______________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________________________ Options ________________________________________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________________________________________________________________ 9 __________________________________________________________________________________________________________________________________________________________________________

______________________________________________________________ Financial Math III—Options ______________________________________________________________

Options are bets on the direction of prices of securities, interest, and currency rates,index levels, and non-financial variables. There are two types of participants in theoptions markets. Speculators (here called by their proper name, but euphemisticallyreferred to as investors in cash markets for stocks or bonds) buy or sell options and waitfor the desired outcome at maturity (a price rise or a rate drop). When their bets proveright, they get a payout; when their bets prove wrong, they do not get a payout or haveto make one themselves. The dealers, or market-makers, do not take outright bets onthe direction of underlying securities. They buy and sell bet tickets, and then manu-facture the payouts for the holders of those tickets. Their risk is that they may over- orunderestimate the cost of the manufacture of the payoffs on the bets that they sold orbought. If they estimate that cost correctly, they always earn a profit, whether prices goup or down, by simply charging a small margin over the cost.This chapter has two central ideas.The first is that there is a big difference between option dealers and insurance sellers

or sports bookmakers. Option dealers buy and sell bets whose payoffs depend on theprices or rates of instruments that themselves can be bought and sold throughout thelife of the option. Insurance companies and bookmakers sell bets on the outcomes offuture events (earthquakes, floods, fires, soccer games). They cannot trade in theseevents; they can only adjust the odds (payoff) or premiums (prices of bets) they offerto reflect the demand and supply for bets. An option dealer who sold a bet that thestock will go up can buy the stock so that if indeed the stock goes up he can profit fromthe appreciation and pass the gain on to the option (bet) buyer. In contrast to a staticcash-and-carry trader who buys the asset once and locks in financing once, the optiondealer does it dynamically. This means that, as the stock rises more, he buys more of it,and when it falls he sells it, expecting not to have to pay on the sold bet. This is what wewill refer to as manufacturing the payoff of the options and what is commonly knownas delta-hedging. The cost of this manufacture depends directly on how volatile theunderlying stock or interest rate is. The more it moves up and down, the more it coststhe dealer to manufacture the desired payoff by the option expiry time. The importantpoint is that the dealer does not speculate on the direction of the stock price.1 If he sellsa bet on the stock going down, he shorts the shares to benefit from the price going downso that he can pass the benefit to the bet buyer. He computes the cost of selling moreshares as they go down and buying them back as they go up. His profit is the margin he

1 Most dealers actually do. The market is so competitive that making profit solely through delta-hedging is very hard. Mostdealers use their knowledge of demand and supply flows to their advantage. This extra profit comes on top of revenue fromcustomer flow.

charges over what he expects the cost of financing the purchases and sales will be.2 Oncehe sells an option, he follows a ‘‘recipe’’ of how to manufacture the payoff. We willpresent the recipe in detail.

The second, related idea is that speculators and dealers in the options markets do notact the same way. The dealer manufactures the payoff by actively trading the underlyingasset while the option is alive. He does not care which way the asset price or rate goes,but he is busy accumulating gains (or losses) which by the expiry date are equal to whathe pays to (receives from) the speculator. The speculator buys (or sells) the bet from thedealer in order to profit from his view, similar to the way the homeowner purchases aninsurance policy in order to ‘‘profit’’ from a fire in his house. The speculator acquires adirectional bet and engages in no further trading. He waits for the outcome to materi-alize. If he is right, he gets (or pays) a payout specified as some formula that depends onthe outcome. For example, if he owns a call option, he gets the excess of the stock priceover some level (strike price).

The central idea here is that the price of the bet the speculator buys from the dealerdoes not depend on the subjective probability or the expected value of the outcome, butonly on the dealer’s cost of manufacture. The latter fluctuates with the perceivedvolatility of the underlying asset. Sometimes bets may seem cheap, sometimes dear,relative to the hoped-for future payoff. When the speculator is right in his predictionabout the event and comes to receive his payoff, the dealer has manufactured thatpayoff through the dynamic hedge strategy (i.e., has accumulated a profit exactlyequal to the promised payoff). When the speculator is wrong and is not owed anypayoff, the dealer has also manufactured that payoff. He has dynamically traded theunderlying asset, in the process losing the entire amount of money received for the betupfront, except for his extra margin (i.e., accruing a zero profit). No matter whathappens, at option expiry or payout time, the dealer has on hand the exact amountthe speculator demands.

Options are redundant securities in the sense that their payoffs can be replicatedthrough dynamic trading strategies. As such, they are not needed. The profit thedealer earns can be thought of as a charge for convenience of having the ready security,so that the speculator does not have to manufacture it himself. This is just like themarkup for ready-made clothes bought off the rack so that we do not have to tailorthem individually. Redundant does not mean useless, however. Options allow verycomplicated risk-sharing schemes for investors. Let us remember that the primaryfunction of securities markets is to channel savings (excess funds) to where they areneeded, whether in productive ventures (ownership shares of small and large businesses)or housing construction (mortgage loans). In order for the original investors (or ontheir behalf, for banks or brokers) to be willing to put their money in those investments,they need to know that they can sell their participation on open markets. That is whywe have stock exchanges and bond markets (see Chapter 1). Having options furtherstrengthens that process. Original investors know that they will be able to sell part oftheir risk in a stock by selling a limited-time bet to give up the upside over a certaindesired level or by buying put options to protect the value of their investments in case


2 The premium (price) of the option reflects his profit and the cost of hedging. This is different from an insurance premium,which reflects the profit plus an unhedgeable bet on the event itself, with the insurer diversifying to improve his chances ofwinning, but nonetheless playing the odds. The option is more akin to a Dell computer. The buyer pays upfront; Dellassembles the computer according to customer specifications and delivers it on the due date.

they go sour. The best analogy is the decision to purchase a house for $1 millionknowing that we can also buy fire and flood insurance, or borrow money against thehouse in case of dire need. Without these possibilities, spending $1 million on a housewould be a much rarer occurrence.This chapter is organized as follows.We start with optionmarkets terminology, payoff

diagrams, and static arbitrage parities. In Section 9.7, we turn to the binomial optionpricing model, the most intuitive way of presenting the dynamic replication argumentand the hedge recipe approach, as trading takes place at discrete times, the way it does inreality. We tie the binomial model to its celebrated continuous-time cousin, the Black–Scholes model. We discuss the residual risks (vega, volatility skew, gamma) of hedgedoption portfolios. At the end of the chapter, we turn to standard interest rate options.These, like swaps within spot and forward markets, represent a huge and fast-growingcomponent of the options markets; yet most readers are only familiar with basic stockoption examples. We also discuss some more common, exotic payoffs.

9.1 CALL AND PUT PAYOFFS AT EXPIRY

Tickets sold by national and state lotteries around the world are bets on a set ofnumbers. The payoff is a fixed monetary amount if the lottery ticket buyer is right inchoosing the right combination. Such bets are called binary or digital. It does not matterhow ‘‘close’’ to the right combination the better is, all that matters is whether he is rightor wrong in guessing five or six numbers.Most common options sold in financial markets work a little differently. The bettor

can also be ‘‘right’’ or ‘‘wrong’’, but the ‘‘more right’’ he is, the bigger his payoff is. Acall option on the price of an asset (e.g., stock) pays on the expiry date the greater ofzero and the difference between that asset’s price and a pre-specified strike price (betlevel). If we want to bet that the price of ABC will go over $60 per share, we buy a callstruck at 60. If the price on the expiry date is 67, we get $7; if the price is 74, we get $14.If the price is below 60, whether at 40 or 50, we get nothing. A put option on a price ofan asset pays on the expiry date the greater of zero and the difference between a pre-specified strike price and that asset’s price. If we want to bet that the price of ABC willgo under $60 per share, we buy a put struck at 60. If the price on the expiry date is 47,we get $13; if the price is 54, we get $6. If the price is above 60, whether at 65 or 80, weget nothing. We graph the payoff the buyer of the option (bet) gets as a function of theunderlying asset’s price in the following payoff diagrams.

Figure 9.1 Payoff on a call and a put at expiry.K ¼ Strike price; S ¼ Spot price at expiry.

Financial Math III—Options 235

Options that pay only at expiry are called European; those that pay prior to and on theexpiry date are called American. Neither notion has any connection to a location.

At any given time, there may be options trading on the same underlying asset price(event), but with different strikes and expiry dates. On the exchanges, options follow acertain schedule of dates and strikes. Over the counter (OTC), they can be arranged forany payoff date and strike (bet level). The intrinsic value of an option is defined as thepayoff the option would have if it were immediately exercisable based on today’s priceof the underlying asset. Options with a positive intrinsic value are called in-the-money;options with no intrinsic value are called out-of-the-money; options whose strike price isequal to the asset price are called at-the-money. Note that options cannot have anegative intrinsic value. If the better is wrong he simply does not exercise his optionand gets no payoff.

Most individual stock options and many others have a physical settlement provision;that is, they are not pure bets, but give the holders (buyers) the right to buy (call) or sell(put) the asset to the option writer (seller) at the strike price on (or prior to) the expirydate. This is tantamount to receiving the payoffs as described above. For example, a 60call, when the price is 67, gives the holder the right to buy the stock from the writer for60 (a private transaction). Once bought, the holder can sell the stock immediately for 67in the spot market (open market), realizing a profit of 7. A 60 put, when the price is 47,gives the holder the right to sell the stock to the writer for 60 (a private transaction). Todeliver, the holder can buy the stock immediately for 47 in the spot market (openmarket), realizing a profit of 13. Many options written on non-price financial variables(stock indices, interest rates, etc.) are settled in cash. The settlement features of optionsare similar to futures that can also have cash settlement or physical settlement. Wealways need to check contract provisions for both, as they may not follow the samerules.

9.2 COMPOSITE PAYOFFS AT EXPIRY

Long and short positions in calls and puts can be combined to achieve a narrowlytailored bet on the range of the asset price in the future. It is important to realize thatspeculators can both buy (hold) and sell (write) options. They can combine long andshort positions. Here are some examples.

Straddles and strangles

Suppose we believe that between now and the expiry date, the price of ABC, currently60, is going to jump dramatically, but we do not know which way. Suppose we buy aput struck at 55 and a call struck at 65. This is called a 55–65 strangle.

On the expiry date, if the price is very low or very high we get a payoff; if the pricestays between 55 and 65 we get nothing. If the price is 30 at expiry, we collect 25 on the55 put and nothing on the 65 call. If the price is 80, we collect nothing on the 55 put and15 on the 65 call. If the price is 62, we get nothing on both options.

Because we bought both options, we incurred an upfront cost of the bet. So our profitis reduced by the total price of the options. An option price is called a premium (likewith insurance).


Next suppose that we believe that on the expiry date the price of ABC is going to stayaround 62. To give ourselves some room for error, we sell a 60–65 strangle (60 put and65 call) and receive premiums for both options. If we are right and the price does not gobelow 60 or above 65, we get to keep the total premium and make no payoff; if we arewrong we have to make a payout on one of the options. The payoffs on the long 55–65strangle and short 60–65 strangle are shown in Figure 9.2.

Figure 9.2 Strangle payoffs at expiry.

A strangle where the call strike and the put strike are the same is called a straddle. Shortstraddles and strangles are bets that the price will not fluctuate much from some level.Long strangles and straddles are bets that the price will move away from the anticipatedlevel.Note that it is not the direction of the position, long or short, which distinguishes the

speculator from the dealer, but it is their actions after the bet is arranged and paid for.The speculator statically waits for the payout; the dealer manufactures it throughdynamic trading.

Spreads and combinations

Another set of popular speculative strategies includes call and put spreads. Thesecombine long and short positions in options of the same type.A long call spread consists of a long low-strike call and a short high-strike call.

Suppose we believe that ABC’s price will rise by expiry, but will not exceed 80. Wecan buy a 65 call and sell an 80 call. Any outcome between 65 and 80 will yield anincreasing payoff; we will have given up any increase at and above 80. If the stock endsup at 76, we will collect 11 from the 65 call and we will pay nothing on the 80 call. If thestock ends up at 88, we will collect 23 from the 65 call, but we will pay 8 on the 80 call,leaving us with 15, the maximum payoff from the strategy reached at 80. Below 65 bothoptions are worthless.A long put spread consists of a long high-strike put and a short low-strike put.

Suppose we believe that ABC’s price will drop by expiry, but not below 35. We canbuy a 55 put and sell a 35 put. Any outcome between 35 and 55 will yield an increasingpayoff; we will have given up any increase below 35. If the stock ends up at 46, we will


collect 9 from the 55 put and we will pay nothing on the 35 put. If the stock ends up at28, we will collect 27 from the 55 put, but we will pay 7 on the 35 put, leaving us with20, the maximum payoff from the strategy reached at 35. Above 55 both options areworthless.

The payoffs on the long 65–80 call spread and short 35–55 put spread are shown inFigure 9.3.

Figure 9.3 Spread payoffs at expiry.

We can package puts and calls into other combinations. If we sell a low-strike put andbuy a high-strike call, we will have no payout over a wide intermediate range of prices(but perhaps a net premium received), we will benefit on the upside, and pay out on thedownside.

Options can also be combined to form calendar spreads. In this case, we buy anoption with one expiry date and sell an option with the same (or different) strike, butwith a different expiry date. For example, if we believe that ABC’s price might first goup but then come down, we can buy a call with short maturity and sell one with a longermaturity. Or if we believe that neither option will ever pay, we may want to sell themore expensive (longer) one and buy the cheaper (shorter) one to pocket the difference.

Options can be combined with long and short positions on the underlying asset andin leveraged proportions.

A buy–write is a strategy where we buy the stock and sell a high strike call. As thestock rises, we are exposed to the possibility that it will rise above the strike of the call.At that level, we will forgo any further appreciation in the stock as we will be forced topay out on the call. If we believe that the stock will appreciate over time but not rapidly,we can opt to sell a string of calls with increasing strikes and maturities. We collect lotsof premium and hope that the options never pay off. This strategy is popular with assetmanagers (e.g., insurance companies) who naturally hold long portfolios of stocks andwant to gain extra income for some extra risk. The strategy is called covered call writing.

In a leveraged buy–write, we buy the stock and sell two calls with the same strike andthe same maturity. If the stock price rises above the bet level we may lose any of theprior appreciation we have gained.

The payoffs on a long, 80-strike buy–write and a 2-to-1 leveraged version of that areshown in Figure 9.4.


Figure 9.4 Buy–write payoffs at expiry (stock at 60).

We can also combine a put with a long stock position to obtain protection against thestock falling. The strategy works for holders of concentrated wealth (e.g., private stocksof wealthy families) who cannot or do not want to sell their holdings. If we are afraidthat ABC’s price may drop below 50, we can buy a 50 put to protect our long stockposition. To reduce the cost of the protection we can sell a 40 put.Options help with investment-timing decisions. Suppose we researched ABC’s stock

and consider it a solid long-term investment. Based on our analysis, we would like tobuy it at 56 or below, but the price has run up to 60. The stock is highly volatile in theshort run. Suppose we sell (write) a 60 put for 5. If the stock continues to go up, at leastwe collected 5; we have to chase the stock up, but we enjoy a defrayment of cost. If thestock goes down to 57, we will be exercised against, having to pay 3. Our net profit is 2and we can use it to buy the stock, effectively paying 55. If the stock goes down evenfurther to 50, we pay out 10. This leaves us with a loss of 5, but we can buy the stock for50, effectively paying 55.The payoffs on a 50 put-protected stock and a long, 40–50 put-protected stock are

shown in Figure 9.5.

Figure 9.5 Payoffs on put-protected stock at expiry (stock at 60).


Binary options

OTC, we can purchase options with fixed monetary payoffs. Suppose we pay 3 to get20, if ABC’s stock goes to or above 70, or we pay 2 to get 20, if ABC’s stock goes downto 50 but not below 40. The payoffs on these two options are in the following diagrams.Binary options are more common with interest rates, where a version of them is called arange.

Figure 9.6 Payoffs on binary options at expiry (stock at 60).

9.3 OPTION VALUES PRIOR TO EXPIRY

In all the above diagrams we showed the payoff of the options on the expiry date. Priorto expiry the option value will have to be higher than the present value (PV) of thepayoff and in most cases even higher than the intrinsic value (equal to the payoffcomputed with today’s stock price). For American options, which can be exercisedimmediately, this should be obvious. We can always exercise the option for its intrinsicvalue, so the value cannot be lower than that. Then there is a possibility of an evengreater payoff. For European options, the argument is a little more subtle. For optionson assets with no intermediate payouts, it can be shown that European calls should notbe exercised early and are thus equally valuable as American calls.3 American puts maybe optimal to be exercised early and so the European options price can be lower than itsAmerican version.

In general, the value of the option prior to expiry can be represented as a line abovethe intrinsic value bound.

Figure 9.7 Value of a call and a put prior to expiry.K ¼ Strike price; S ¼ Spot price now.


3 For example, an American call on a stock that pays no dividends should not be exercised early and is thus just as valuable asa European one.

Option prices observe certain obvious arbitrage rules. A higher strike call will always beworth less than a lower strike call with the same expiry. If ABC’s stock is at 60, then a70 call will be worth more than an 80 call because the 70 call will always pay more thanan 80 call. If the stock ends up at 75, the 70 call will pay 5 while the 80 call will paynothing. The two will pay the same only if they both pay nothing. A lower strike putwill be always worth less than a higher strike put with the same maturity. The reasoningis analogous to the call case.An option with a longer maturity will always be worth at least as much as an

otherwise identical option with a shorter maturity. If the options have the samestrikes, then their intrinsic values will be the same, but the longer option will allowmore time for the underlying asset to move to generate a higher payoff. The analogy isthe insurance policy comparison between short- and long-term coverage. Since long-term coverage allows for a greater possibility of a payoff, it will cost more. Insurancehas a similarly asymmetric payoff. It pays nothing if the desired event does not takeplace, no matter how ‘‘close’’ to it we got; on the other side, the payoff depends on howdeep in-the-money we get. It is only one side of the probability distribution thatdetermines the value of the policy.We will discover a few more arbitrage rules later.

9.4 OPTIONS, FORWARDS, AND RISK SHARING

Suppose we buy a European call and sell a European put on the same asset. Supposethat we also search among all possible strike prices and find one strike price K , suchthat the premium on the call struck at K that we buy is exactly paid for by the premiumon the put struck at K that we sell. Let us examine the payoff of our strategy at expiryfor physical-settle options.If S � K , then the call option is in-the-money and we buy the stock for the amount

K. If S � K , then the put option is in-the-money. We are exercised against, and we buythe stock for the amount K . By buying the call and selling the put, we are in effectagreeing to buy the stock for the amount K on the expiry date. In a cash settle case, thismeans that no matter whether the stock price is above or below the strike price, ourpayoff is S � K on the expiry date.Our strategy is equivalent to entering a long forward on the stock. Net, we pay

nothing today, and on a future date we deliver the sum of money K for one share ofthe stock. The forward is on-market (zero PV upfront).If we were slightly less careful in our search for the perfect strike and, instead, picked

one at random, but made sure that the strike on the long call is the same as the strike onthe short put, then we would have in effect entered into an off-market forward with themark-to-market value (PV) equal to the difference between the premium on the call andthat on the put.

Call � Put ¼ Forward

We can think of the forward value of the stock as the median separating two possibilityregions for the future stock price. The long call covers the part of the region to the rightof the strike (i.e., with stock values greater than the strike). The short put covers thepart of the region to the left of the strike (i.e., with stock values lower than the strike).Traders can synthesize forwards from options or they can enter into forwards and


synthesize options by selling off the undesirable probability regions. For example, tosynthesize a call a dealer may enter into a forward and buy a put to offset the short putimplicit in the forward. To synthesize a put, a dealer may enter into a short forward andbuy a call to offset the short call implicit in the short forward.

This is an advanced way of risk arbitrage. The prices of calls and puts have to be inline with on- and off-market forwards and futures. The arbitrage is executable bycombining futures and options on the same underlying asset and choosing to buy theside that is cheaper relative to the other.

This is also an advanced way of broad risk sharing. When a stock is bought in an IPOproviding capital to a growing business, the buyer may not think much of options andforwards. He does, however, appreciate the existence of a secondary market for stocks(stock exchange), so that he can sell the stock when he no longer wants to bear the riskof the stock. The person he sells to may, however, be an option player who wants thestock, but only for a certain amount of time or only in a certain scenario. The fact thathe can customize his participation in the stock may be the main reason that hepurchases the stock. He does not buy another stock that does not have optionstrading on it, because that would force him into an all-or-nothing risk.

9.5 CURRENCY OPTIONS

Options on currencies work the same way as options on other assets. However, just likein spot and forward foreign exchange (FX) rates, we need to be careful to distinguishthe pricing currency (numerator) and the priced one (denominator). With other assets,this is natural: the underlying price and the strike price of a stock option are expressedin dollars per share of ABC’s stock. We are never interested in inverting the price or thestrike to know how many shares per dollar we can buy. With currencies we often do.Because of that, each currency option (i.e., its strike rate) can be defined in two differentways (e.g., in yen per dollar and in dollar per yen). To keep things straight, one shouldthink of the denominator currency as the underlying asset and the numerator currencyas a price unit.

Let us consider a call option on the U.S. dollar with a strike of 110 Japanese yen perdollar (think of the dollar as the underlying asset) with one call covering $1,000 (thinkof dollars as shares and each call is on 1,000 shares). If the dollar’s price rises aboveY¼110, the holder gets a payoff, otherwise not. The payoff, as with any call is equal to thedifference between the spot price of the underlying asset S (i.e., the spot FX rate in Y¼/$)on the expiry date minus the strike K , or nothing, times the number of units of theunderlying asset (principal amount or size). That is, just like with any call, it is:

CallJPY=USD ¼ SizeUSD �MaxðSJPY=USD � KJPY=USD; 0Þwhere both S and K are in Y¼/$. The payoff is denominated in Y¼. If the spot FX rate atexpiry is Y¼117, we get Y¼7 times the 1,000 unit size, or Y¼7,000, equivalent to$59.829 059 83 at the Y¼/$117 FX rate; if the spot FX rate at expiry is Y¼102, we getnothing. A physical-settle version of this option would be the right to buy 1,000 dollarsfor Y¼110 a piece, which would be exercised only if the spot value of the dollar is greaterthan Y¼110. But the right to buy the dollar is automatically equivalent to the right to sellthe yen.


The call on the dollar struck at Y¼/$110 is also a put on the yen struck at 1/110 ¼ $/Y¼0.009 090 91. If the size of the call is 1,000 dollars, then the size of the put is Y¼110,000(converting using the strike FX rate):

CallUSD=JPY ¼ SizeJPY �MaxðKUSD=JPY � SUSD=JPY ; 0ÞWhen the value of the yen goes down to $0.008 547 01 (i.e., the spot FX rate moves toY¼/$117), we get a payout of 0.009 090 91�0.008 547 01 ¼ $0.000 543 90 per unit of yen.This times the size of the option, 110,000 units of yen, gives us the total payoff of$59.829 059 83, which is the same as Y¼7,000 at the Y¼/$117 FX rate. When the spot FXrate goes up to $/Y¼0.009 803 92 (i.e., Y¼/$102), we get nothing.

Call on currency1 ¼ Put on currency2Size converted at the strike FX rate (strike rate inverted)

This rule is true for all currency options, standard or not.

9.6 OPTIONS ON NON-PRICE VARIABLES

Options can be written on any variable, not just a price of some asset. Suppose we writea put option on the temperature reading in Paris on July 15, 2008 with a striketemperature of 23�C. We need to define how the outcome of the event will be translatedinto a monetary payoff. With stocks or currencies, this is automatic. Once we know thesize of the option, say 100 shares or £62,500, the payoff is equal to the price differenceper unit times the size. With options written on non-price variables, we have to define anumber that translates the units of the non-price variable into money. Once we specify amultiplier of c¼ 500 per one degree centigrade, the definition of our option is complete. Ifthe temperature in Paris on the expiry date is 17�C, our put option pays 23� 17 ¼ 6times 500, or c¼ 3,000. If the temperature is 26�C, the put pays nothing.The first, most common example of a non-price variable is a stock index. A stock

index is not a price of anything, but a normalized number designed to track thepercentage changes in a particularly defined basket of stocks. The basket changesover time as some stocks come in and some are removed. Consider the Nikkei 225index. In order to define a payoff of an option we need to translate the index points intoyen. For example, we may specify that an 11,000 call will pay the difference between theindex value and strike level times Y¼10,000, or 0, whichever is greater. So if the index hits11,078.23 on the expiry date of the call, the holder would get:

ð11,078:23� 11,000Þ �Y¼10; 000 ¼ Y¼782,300

Note that the multiplier need not be specified in what seems natural. Suppose a U.S.investor wants exposure to the Japanese stock market, but does not want to bearcurrency risk. We could define the multiplier in dollars per Nikkei point (e.g., $250per point). The holder would then get:

ð11,078; 23� 11,000Þ � $250 ¼ $19,557:50

(For a dealer, this dollar option is much harder to hedge. The yen option is hedged bybuying stocks in the right proportions, using the multiplier to define the yen amounts to


be spent on shares. The dollar option involves additional currency exposure as yengains and losses on the stocks have to be translated into dollars at a fictitious one-for-one fixed rate.)

Another non-price-based option example is an interest rate. Options can be writtenon the price of a bond, in which case we only need to specify the face value of the bondas the size. But options can also be written directly on the interest rate, whether spot orforward. The rate cannot be bought or sold (only an instrument whose value dependson it), so we cannot specify a ‘‘size’’. But we can specify a multiplier, say as $100 per1 basis point (bp) of the difference between the underlying rate and the strike rate.(Hedging may be difficult. The option fixes a linear yield–payout relationship; the price–yield relationship for the hedged instrument is non-linear, so a $1 price change does nottranslate to a 1% yield change.)

Sometimes an interest-rate-based option’s multiplier is implicit. A cap is a multi-period ‘‘call’’ option on a short-term rate which consists of several caplets. Eachcaplet provides a payment for one interest accrual period equal to 0 or the differencebetween the rate and a strike, whichever is greater, times the notional principal amounttimes the appropriate day-count. A 5-year, $100 million 4% cap on the 3-month U.S.dollar LIBOR (London interbank offered rate) consists of 20 caplets, one for eachsubsequent 3-month period. Caplet 9’s payout will be based on the greater of zeroand the difference between the 3-month LIBOR 2 years from today and 4%, andwill take place 3 months later (i.e., in 2 years and 3 months); this is designed tomimic the way swaps and floating rate bonds pay. The multiplier for each caplet isequal to $100 million times the day-count for the relevant 3-month interest period (e.g.,Act/360). Suppose 2 years from today LIBOR sets at 4.34%. The payout on the cap forthat interest period assuming it has 92 days will be:

$100,000,000� ð4:34%� 4%Þ � 92=360 ¼ $86,888:89

The multiplier on the cap changes slightly for each period as the day-count fractionchanges.

9.7 BINOMIAL OPTIONS PRICING

The option premium charged by a dealer reflects his cost of manufacturing the payoff.The dealer sells (or buys) the option, then borrows or lends money, and takes a partialposition in the underlying asset. By the expiry time, his hedge is worth exactly the sameas the payoff on the option he owes or receives. We will illustrate the mechanics ofpayoff manufacturing with increasingly more revealing examples of binomial trees.4 Allexamples use stocks, but are equally applicable to other traded assets.

One-step examples

We use the following assumptions for Examples 1–3. The underlying stock sells cur-rently for S ¼ $50 a share. The expiry of the option is 1 year from today (or one periodwith no trading in the underlying between now and expiry). The dealer sells the option,


4 The examples follow the notation and exposition used by many option textbooks, notably the ‘‘bible’’ of John C. Cox andMark Rubinstein, Option Markets, 1985, Prentice Hall, Englewood Cliffs, NJ, and Martin Baxter and Andrew Rennie,Financial Calculus, 1996, Cambridge University Press, Cambridge, U.K.

collects the premium, and then follows a set of instructions. For Examples 1, 2a, and 3a,we also assume that the dealer can borrow or lend money at no interest. On the expirydate, one period from today, the stock can take on two values Sup ¼ $70 or Sdn ¼ $20.The dealer believes that the up probability is 1

4and the down probability is 3

4. He takes

the following steps:

(1) Given the potential stock outcomes Sup ¼ $70 or Sdn ¼ $20 for the up and downstates tomorrow and given today’s stock price of S ¼ $50, the dealer computes anumber:

q ¼ S � Sdn

Sup � Sdn

We will refer to q as the risk-neutral5 probability of the up state and to 1� q as therisk-neutral probability of the down state. These are the only probability-likenumbers that the dealer uses in his weighted average calculations, not his subjectivebeliefs 1

4and 3

4(he could easily be wrong). The risk-neutral probability q has no

meaning outside the context of this six-step procedure (i.e., it is not a real prob-ability of anything). In our examples, we compute q to be:

q ¼ 50� 20

70� 20¼ 3

5¼ 0:60

(2) Given the strike level K and the potential stock outcomes Sup ¼ $70 or Sdn ¼ $20for the up and down states, he assigns the call payoffs Cup and Cdn, or put payoffsPup or Pdn, for the corresponding states of nature in the expiry period.

(3) He computes the premium on the option by taking the average of the future optionoutcomes weighted by the risk-neutral probabilities of the states; that is:

C ¼ qCup þ ð1� qÞCdn or P ¼ qPup þ ð1� qÞPdn

(4) He computes a hedge number

D ¼ Cup � Cdn

Sup � Sdnor D ¼ Pup � Pdn

Sup � Sdn

which tells him how many shares of stock he needs to hold (buy or sell) today.(5) He buys/sells the prescribed number of shares by paying/receiving D� S. He uses

the collected premium in the purchase or sale. If necessary, he borrows/lends D� Sminus the option premium, so that his cash position today is 0.

(6) He liquidates his hedge one period from today when the state of nature is revealed(i.e., the stock either goes up or down). He uses the proceeds to settle his borrow-ing/lending and to pay the agreed-on payoff to the option buyer.

We will show that if he faithfully follows steps 1–5, then in step 6 he will always have onhand the exact amount of money demanded by the option holder, no matter whathappens to the stock price. He will not have used his subjective beliefs to gamble onthe direction of the stock.


5 The word ‘‘risk-neutral’’ reflects a complicated mathematical concept of a probability measure change. We do not need tounderstand here what that is about. We simply blindly compute a number that we call q, which happens to resemble aprobability. We will use it later in our mechanistic recipe.

In our illustrations, we will place all the computed numbers diagrams with nodes likethis. Next to today’s stock price of S ¼ 50, we will show all the numbers from steps 1–5(i.e., q, D, the option premium C or P, the cost of shares D� S, and the amount ofborrowing/lending D� S � C or D� S � P). Next to the potential future stock pricesSup and Sdn, we will show the corresponding value of the option Cup (or Pup) and Cdn (orPdn), the value of the stock position held from the previous step D� Sup and D� Sdn,and the cash position carried over from the previous step.

Figure 9.8 A binomial node. Current stock price S ¼ 50.

Example 1 (Binary lottery, zero interest rate) John Dealer sells a binary ‘‘call’’ optionon the stock that pays $10 if the stock ends up at or above $60 or nothing if it ends upbelow $60 one period from today. John’s calculations are shown in Figure 9.9.

Figure 9.9 A binary call with payoff C ¼ 10 if S > 60, otherwise zero.

The risk-neutral probability q ¼ 0:60. The call payoffs 1 year from today are 10 if thestock is at 70 or 0 if the stock is at 20. So he sells the call for:

C ¼ 0:60ð$10Þ þ 0:40ð$0Þ ¼ $6

Given his hedge ratio:

D ¼ 10� 0

70� 20¼ 1

5¼ 0:20

he buys 0.20 shares for 0.20� 50, or $10. Since he collected only $6 for the option, heborrows 10� 6 ¼ $4.

One period later, if the stock is at $70, his stock position is worth 0:20� 70 ¼ $14. Heliquidates it, pays $10 on the option, and repays the borrowing of $4. If the stock is at$20, his stock position is 0:20� 20 ¼ $4. He liquidates it, pays nothing on the option,and repays his borrowing of $4. Collecting the premium of $6 on day 1 has allowed


John to manufacture the payoff he is obligated to make irrespective of whether thestock goes up or down.

Example 2a (call struck at 55, zero interest rate) John Dealer sells a standard calloption on the stock struck at $55. At expiry, the call pays the value of the stock (i.e., Sup

or Sdn) minus the strike (K ¼ 55) if the stock ends up at or above $55 or nothing if itends up below $55 one period from today. John’s calculations are as follows.

Figure 9.10 A call struck at 55. Payoff C ¼ maxðS � 55, 0Þ. Zero interest rate.

The call payoffs are 15 if the stock is at 70 or 0 if the stock is at 20. So he sells the callfor:

C ¼ 0:60ð$15Þ þ 0:40ð$0Þ ¼ $9


D15� 0

70� 20¼ 3

10¼ 0:30

he buys 0.30 shares for 0.30� 50 or $15. Since he collected only $9 for the option, heborrows 15� 9 ¼ $6.One period later, if the stock is at $70, his stock position is worth 0:30� 70 ¼ $21. He

liquidates it, pays $15 on the option, and repays the borrowing of $6. If the stock is at$20, his stock position is 0:30� 20 ¼ $6. He liquidates it, pays nothing on the option,and repays his borrowing of $6.

Example 3a (put struck at 55, zero interest rate) John Dealer sells a standard putoption on the stock struck at $55. At expiry, the put pays the strike (K ¼ 55) minus thestock value (Sup or Sdn) if the stock ends up at or below $55 or nothing if it ends upabove $55 one period from today. John’s calculations are as follows.

Figure 9.11 A put struck at 55. Payoff P ¼ maxðS � 55, 0Þ. Zero interest rate.


The put payoffs are 0 if the stock is at 70 or 35 if the stock is at 20. So he sells the putfor:

P ¼ 0:60ð$0Þ þ 0:40ð$35Þ ¼ $14


D ¼ 0� 35

70� 20¼ � 7

10¼ �0:70

he shorts 0.70 shares to collect 0.70� 50, or $35. Since he collected $14 for the option,he places the combined proceeds 35þ 14 ¼ $49 in a deposit (i.e., lends).

One period later, if the stock is at $70, his stock position is worth�0:70� 70 ¼ �$49. He liquidates it. He uses the $49 from the deposit to buy thestock back and return it to the lender. He pays nothing on the option. If the stock isat $20, his stock position is �0:70� 20 ¼ �$14. He uses the $49 from the deposit toliquidate the stock position ($14) and to pay $35 on the option.

Let us make a few observations. Once the payoff on the option is defined, the rest is amechanical adherence to a recipe. The recipe covers all potential payoff structures—binary, standard, or any other exotic—as well as both puts and calls and both boughtand sold options. The actions for the dealer who buys the option, instead of selling,would be analogous. They are completely determined by his hedge ratio D. A positivedelta means long stock; a negative one means short stock. The borrowing and lendingsimply balances the cash position resulting from the price of the D amount of stock andthe premium on the option.

Let us now demonstrate that the recipe works with only slight modifications when wedo not make the unrealistic assumption that the financing interest rate is 0. Here are theamendments:

(1) The formula for q is changed by replacing today’s S with its future value equivalent(i.e., the forward). Recall that the forward is equal to the value of S multiplied by afuture value factor, equal to 1 plus the interest rate r for 1 year. For fractions of ayear, or special compounding and day-count conventions, it needs to be amendedappropriately. For an annual period it is:

q ¼ Sð1þ rÞ � Sdn

Sup � Sdn

In our example, we assume that r ¼ 10% and compute q to be:

q ¼ 50ð1:1Þ � 20

70� 20¼ 7

10¼ 0:70

(2) No change.(3) We compute the premium on the option by taking the average of the future option

outcomes weighted by the risk-neutral probabilities of the states, present-valued totoday; that is:

C ¼ 1

1þ r½qCup þ ð1� qÞCdn� or P ¼ 1

1þ r½qPup þ ð1� qÞPdn�

(4) through (6) No change, but we have to remember about interest paid or earned onborrowing or lending when carrying over the cash position from the previous step.


Let us repeat the standard call and put example with a financing cost of 10%.

Example 2b (call struck at 55, 10% interest rate) John Dealer sells a standard calloption on the stock struck at $55. At expiry, the call pays the value of the stock (i.e., Sup

or Sdn) minus the strike (K ¼ 55) if the stock ends up at or above $55 or nothing if itends up below $55 one period from today. John’s calculations are as follows.

Figure 9.12 A call struck at 55. Payoff C ¼ maxðS � 55, 0Þ. 10% interest rate.

The call payoffs are still 15 if the stock is at 70 or 0 if the stock is at 20. But he sells thecall for:

C ¼ 1

1:1½0:70ð$15Þ þ 0:30ð$0Þ� ¼ $9

6

11¼ $9:54545


D ¼ 15� 0

70� 20¼ 3

10¼ 0:30

he buys 0.30 shares for 0.30� 50, or $15. Since he collected only $9 611for the option, he

borrows 15� 9 611¼ $5 5

11.

One period later, if the stock is at $70, his stock position is worth 0:30� 70 ¼ $21. Heliquidates it, pays $15 on the option, and repays the borrowing, which has by nowaccrued to $6 (5 5

11� 1:1) at the 10% interest rate. If the stock is at $20, his stock

position is 0:30� 20 ¼ $6. He liquidates it, pays nothing on the option, and repayshis borrowing and interest of $6.

Example 3b (put struck at 55, 10% interest rate) John Dealer sells a standard putoption on the stock struck at $55. At expiry, the put pays the strike (K ¼ 55) minus thestock value (Sup or Sdn) if the stock ends up at or below $55 or nothing if it ends upabove $55 one period from today. John’s calculations are as follows.

Figure 9.13 A put struck at 55. Payoff P ¼ maxð55� S, 0Þ. Zero interest rate.


The put payoffs are unchanged: 0 if the stock is at 70 or 35 if the stock is at 20. He sellsthe put for:

P ¼ 1

1:1½0:70ð$0Þ þ 0:30ð$35Þ� ¼ $9

6

11¼ $9:54545


D ¼ 0� 35

70� 20¼ � 7

10¼ �0:70

he shorts 0.70 shares to collect 0.70� 50, or $35. Since he collected $9 611for the option,

he places the combined proceeds 35þ 9 611¼ $44 6

11in a deposit (i.e., he lends).

One period later, if the stock is at $70, his stock position is worth�0:70� 70 ¼ �$49. He liquidates it. He uses the $49 (44 6

11� 1:1) from the deposit,

which has accrued interest in the meantime, to buy the stock back and return it to thelender. He pays nothing on the option. If the stock is at $20, his stock position is�0:70� 20 ¼ �$14. He uses the $49 from the deposit and interest to liquidate thestock position ($14) and to pay $35 on the option.

Let us make a few more observations. First, the hedge ratios are the same as in thezero interest case, but the borrowing/lending grows period to period and makes adifference in the final apportioning of the proceeds at expiry.

Second, a positive interest rate raised the price of the call and lowered the price of theput (Examples 2b and 3b relative to 2a and 3a). This is because a call seller borrowsmoney to buy stock thereby incurring a cost, while a put seller lends money aftershorting a stock accruing interest. So the cost of manufacturing the final payoff in-creases for the short-call hedger and decreases for the short-put hedger.

Third, the price of the call and the put in our example was the same (Examples 2band 3b). This was not a coincidence. The strike price on both options was equal to $55.This is the forward price of the stock for delivery on the expiry date, equal to the spotprice of the stock $50 times a future value factor reflecting the cost of carry (i.e.,50� (1þ 0:10) ¼ $55).

This confirms our prior assertion, before we knew anything about option pricing,that a call and a put struck at a forward will have the same cost, so that we canmanufacture the forward by buying a call and selling a put struck at the forward price.

Let us further show that the last property will hold no matter how volatile the stock isbetween now and expiry. Suppose that, instead of potential outcomes of $70 or $20, thestock is perceived to have potential outcomes of Sup ¼ $80 or Sdn ¼ $15. The stock ismore volatile and is thus riskier. We follow our recipe using an interest rate of 10%.The forward value of the stock is still the same $55.

Example 2c (call struck at 55, 10% interest rate) John Dealer’s calculations are asfollows.

Figure 9.14 A call struck at 55. Payoff C ¼ maxðS � 55, 0Þ. 10% interest rate. Higher volatility.


He sells the option for $13.986, a lot more than before, to reflect the increasedexpected value of the payoff.

Example 3c (put struck at 55, 10% interest rate) John Dealer’s calculations are asfollows.

Figure 9.15 A put struck at 55. Payoff P ¼ maxð55� S, 0Þ. 10% interest rate. Higher volatility.

Again, he sells the option for $13.986, a lot more than before (Examples 2c and 3crelative to 2b and 3b), to reflect the increased expected value of the payoff.In both examples, he sells the call for the same price as the put (Examples 2c and 3c).

It will always be true that if the perceived riskiness of the underlying asset increases,both calls and puts will increase in value, but the price of a call struck at a forward willalways be equal to the price of a put struck at a forward. This is because the forwarddoes not have anything to do with the volatility of the stock, it simply reflects the cost ofcarry. A long-call–short-put position, equivalent to the forward, must carry a net zeropremium (an on-market forward costs nothing to enter into).What we have also shown is that, while the subjective probabilities of the stock

outcomes are irrelevant, the volatility, or the potential dispersion of the outcomes, isnot. The more volatile the stock is, the higher are the premiums on standard calls andputs (Examples 2c and 3c relative to 2b and 3b). This reflects the asymmetric nature oftheir payoffs. A more volatile stock means that the payoff when the option is in-the-money is likely to be larger, while when the option is out-of-the-money the payoff is stillthe same constant zero. Thus the expected value of the payoff is higher if the volatility ishigher.

A multi-step example

Let us now demonstrate the full dynamic process of hedging an option (i.e., manufac-turing its payoff). We consider a put struck at K ¼ 54, an interest rate r ¼ 2:6% perperiod (i.e., already de-compounded), and a stock price currently at 50, and follow thedynamics shown in Figure 9.16:


Figure 9.16 A binomial tree. Current stock price S ¼ 50.

For example, an option with 3 months to expiry might be divided into monthly steps.Over the first month, the stock can go up to 60 or down to 35 (the actual probability ofeach step is irrelevant). If the stock went down to 35 during the first month, then it cango up to 45 or down to 20 over the second month, etc. For clarity, we will drop thearrows for the rest of the exposition.

We follow the same logic as used in the one-step examples for each subtree. We firstcompute the risk-neutral probability:

q ¼ Sð1þ rÞ � Sdn

Sup � Sdn

for all subtrees. For example, for the subtree emanating from the 45 point, we have:

q ¼ 45ð1þ 0:026Þ � 30

55� 30¼ 0:6468

We also determine the payoff of the option at expiry. For example, when the stock priceis 45, the payoff would be 54� 45 ¼ $9. We place the q’s and the final payoffs in thediagram.

Figure 9.17 A binomial tree. Current stock price S ¼ 50. Put payoff P ¼ maxð54� S, 0Þ. Com-puted q’s and final payoffs.


As in the one-step examples, we sweep through the tree backward to determine thepremium on the option today. We use the same equation as before for each node:

C ¼ 1

1þ r½qCup þ ð1� qÞCdn� or P ¼ 1

1þ r½qPup þ ð1� qÞPdn�

We start with the second-to-last date and consider the subtrees emanating from all threepoints. We compute put values for all three states: 70, 45, and 20. For the $45 state wecompute:

P ¼ 1

1þ 0:026½0:6468 � 0þ ð1� 0:6468Þ � 24� ¼ 8:2620

For the $20 state we compute:

P ¼ 1

1þ 0:026½0:6208 � 24þ ð1� 0:6208Þ � 49� ¼ 32:6316

We place the values on the tree diagram. We go to one date before the one justcomputed and calculate the put values for each node (60 and 35) on this date, usingthe same equation linking a node on a given date to two future nodes. For example, forthe $35 state we get:

P ¼ 1

1þ 0:026½0:6364 � 8:2620þ ð1� 0:6364Þ � 32:6316� ¼ 16:6889

We proceed recursively like this until we obtain today’s value of the put P ¼ $7:3881.

Figure 9.18 A binomial tree for a put struck at 54. Computed q’s, final payoffs and premiums.

Next, we demonstrate that no matter which route the stock price takes between todayand 3 months from today, the hedge will work perfectly.For each node, we compute the hedge number:

D ¼ Cup � Cdn

Sup � Sdnor D ¼ Pup � Pdn

Sup � Sdn

which tells us how many shares we should hold at that node. In our put example, alldeltas will be negative or 0 to reflect the fact that we will short shares. For example, for


the $35 state 1 month from today the delta is:

D ¼ 8:2620� 32:6316

45� 20¼ �0:9748

Again we place all the deltas in the diagram.

Figure 9.19 A binomial tree for a put struck at 54. Added D’s for all nodes.

Depending on the route the stock takes, all trades are now determined by the differencesbetween deltas at subsequent nodes. The lending amounts are determined too by thecash position at each node. Let us go through the tree forward, following one hypo-thetical path.

Suppose the stock price from today’s level of $50 goes down to $35 1 month fromtoday, then to $45 2 months from today, and ends up at $30 3 months from today. Inorder to hedge our position, we are required to short 0.5588 shares today. This willresult in proceeds of 0:5588� 50 ¼ $27:9406. We deposit that and the premium re-ceived from selling the put (i.e., a total of 27:9406þ 7:3881 ¼ $35:3287) in anaccount earning 2.6% per month.

Figure 9.20 Today’s node with S ¼ 50, P ¼ 7:3881, D ¼ �0:5588. Proceeds from short sale27.9406. Total lending 35.3287.

The following month the price goes down to $35. Based on our new delta of �0:9748,we need to short an additional 0:9748� 0:5588 ¼ 0:4159 shares. This results inproceeds of 0:4159� 35 ¼ $14:5590. Meanwhile, our prior lending accrued to35.3287� (1þ 0.026) ¼ 36.2473. We relend the sum of the two (i.e., 14.5590þ36.2473 ¼ $50.8063) for another month at 2.6%. Note that the borrowing/lendingamount can also be found by subtracting the put value at a node, 16.6889, from thevalue of the share holding, �0:9748� 35 ¼ $� 34:1174 (i.e., �34:1174� 16:6889 ¼$� 50:8063Þ:


Figure 9.21 Month 1’s node with S ¼ 50, D ¼ �0:9748.

The following month, the stock price increases to $45. Based on our new delta of�0:9600, we need to buy back 0:9748� 0:9600 ¼ 0:0148 shares. This costs us0.0148� 45 ¼ $0.6653. We take that amount from the maturing deposit which hasaccrued to 50.8063� (1þ 0.026) ¼ $52.1273. We relend the remainder 52:1273�0:6653 ¼ $51:4620 for another month at 2.6%. Again, the borrowing/lending amountcan be found by subtracting the put value at a node, 8.2620, from the value of theshareholding, �0:9600� 45 ¼ $� 43:2000 (i.e., �43:2000� 82620 ¼ $� 51:4620).

Figure 9.22 Month 2’s node with S ¼ 45, D ¼ �0:9600.

We proceed to the final step. The stock goes down to $30. We collect the deposit withaccrued interest (i.e. 51.4620� (1þ 0.026) ¼ $52.8000). We buy back the shorted sharesfor 0.9600� 30 ¼ $28.8000 and pay $24 to the put holder. We are left with no stockposition, no borrowing or lending position, and our put obligation is satisfied.

Figure 9.23 Final month’s node with S ¼ 30. Payoff of 24.

We can trace any other path through the tree to see that the result would be identical:we would end up with no stock, no cash, and we would have made a payout on the put,if any was required. The summary of all the calculations is portrayed in the completeddiagram.

Figure 9.24 A complete binomial tree for a put struck at 54, including premiums, D’s, totalsshorted and totals lent.


The procedure of computing the q’s, sweeping backward to get the option valueupfront, and sweeping forward to compute the required hedges and borrowing/lending positions works for all standard calls and puts, digital options, barrieroptions, American exercise style, and many other options. In all of these cases, theonly thing that changes is the recursive computations of the option value during thebackward sweep. For example, for American options that can be exercised early, wehave to amend the option value for any given node to see if the immediate exercise valueis not greater than the unexercised value; that is:

C ¼ Max

�1

1þ r½qCup þ ð1� qÞCdn�;S � K

�

or

P ¼ Max

�1

1þ r½qPup þ ð1� qÞPdn�;K � S

�

This is very easy to implement in any computer code or spreadsheet.

Black–Scholes

The well-known Black–Scholes6 equation for calls and puts is a continuous general-ization of the binomial approach. There are at least two improvements there: first, thestock price, looking forward from one date to the next, can take on a continuum ofvalues, not just two; and, second, there is a continuum of dates, not just month tomonth or day to day. The equation computes the present value of a hedge strategywhere the rebalancing occurs instant by instant and over minute price changes. Theoverriding principle of payoff manufacturing remains the same. An option payoff isreplicated by a position in a stock combined with borrowing or lending. The stock andbond position is adjusted continuously and for infinitesimal value changes. The value ofthe option today is equal to the cash required to start this dynamic hedge process. Foroptions on a non-dividend-paying stock, it is equal to:

C ¼ SNðd1Þ � Ke�rTNðd2ÞP ¼ Ke�rTNð�d2Þ � SNð�d1Þ

where d1 ¼lnðS=KÞ þ ðrþ �2=2ÞT

�ffiffiffiffiT

p , d2 ¼ d1 � �ffiffiffiffiT

p, r is a continuously compounded

interest rate, and T is time to maturity (in years). The formula provides an explicit linkbetween the annual volatility of the stock return � and the value of the option. In theBlack–Scholes model, the continuously compounded return on the stock over an in-finitesimal interval dt is assumed to be normally distributed. What this means is that,instead of the stock price taking on potentially only two values when moving from timet to time tþ dt, the stock can take on a continuum of values such that the continuouslycompounded return on the stock over that interval, lnðStþdt=StÞ, has a standard devia-tion of �

ffiffiffiffiffidt

p. This is portrayed in Figure 9.25.


6 Fischer Black and Myron Scholes, ‘‘The pricing of options and corporate liabilities’’, Journal of Political Economy, 81,637–659, May/June, 1973. The model is also attributed to Robert C. Merton, ‘‘Theory of rational option pricing’’, BellJournal of Economics and Management Science, 4, 141–183, Spring, 1973.

Figure 9.25 The probability density for stock price outcomes under log-normal distribution.

The probability mass curve in the graph is not the normal bell curve but rather that of alog-normal. The minimal stock price is 0, the maximal price is1. The percentage returnis normal and would have a bell-shaped curve.The inclusion of price volatility in arriving at the value of the option was implicit in

our binomial trees, where the width of each branch depended on the variability of thestock. If we set:

u ¼ e�ffiffiffiffiDt

p; d ¼ 1=u; q ¼ erDt � d

u� d

then it is easy to show that our binomial model with a time step of Dt is just a discreteapproximation of and will converge to the Black–Scholes equation7 as we shorten Dt(i.e., increase the number of rehedging times in the tree). That is why we chose topresent the binomial approach to option pricing first, rather than go the continuousequation route. The discrete model is intuitive and more general, as it allows Americanexercise options to be valued; the continuous approach relies on a stochastic calculusargument. The two can be made equivalent to each other. That is, they come up withthe same answer to the premium and hedge ratios.

Dividends

The inclusion of dividends in option pricing is straightforward, assuming that dividendsare a known cash payout between now and the option expiry. Their effect on optionprices should be intuitive. Because they are an outflow of value from the stock, theyreduce the potential future outcomes for the stock. As such, their impact is to decreasethe call values and to increase the put values. This can also be argued by considering thedelta hedge. The call writer shorts less of the stock, and the put writer buys more of


7 This is only one of many possible ways of making the binomial tree converge to the Black–Scholes equation. For adiscussion, see John C. Hull, Options, Futures, and Other Derivatives (4th edn), 2000, Prentice Hall, Englewood Cliffs, NJ.

the stock, because when they adjust the hedge the stock price will have been reduced bythe amount of the dividends, whether the stock has gone up or down.

There are two ways to correct option valuation for dividends, depending on whethertheir amount is known in dollars or as a percentage of the stock price (analogously tothe forward discussion in Chapter 6). Consider dividends paid at a constant continuousrate �. In the Black–Scholes model, the inclusion of dividends is accomplished8 bymultiplying each occurrence of the stock price S by the continuous compoundingterm e��T . In the binomial model, the correction is the same. Each node’s stockvalue is reduced by the amount of accrued dividends. We change the definitions ofthe upstate and the downstate to u ¼ e��Dtþ�

ffiffiffiffiDt

p, d ¼ e��Dt��

ffiffiffiffiDt

p.

9.8 RESIDUAL RISK OF OPTIONS: VOLATILITY

Can anything go wrong with the hedge? If the dealer adheres strictly to the algorithmand as long as the stock follows one of the considered paths, then the answer is no.And, if the Black–Scholes model considers a continuum of paths, doesn’t the algorithmconsider all possible paths? Unfortunately, the answer is no.

The range of possible outcomes considered is determined by the assumed volatility ofthe stock. Volatility is the square root of variance. Variance is the expected value ofsquared deviations of the stock’s return from the mean return over a given period. TheBlack–Scholes model requires volatility as an input. It then considers all the stock pricepaths that are within a certain range bounded by the volatility. That is, stock pricemovements period to period are restricted not to jump discontinuously. Graphically,this can be portrayed as all the paths within an expanding cylinder of outcomes, withthe greatest density of paths close to the center. The edges of the cylinder are notbinding, but the probability of outcomes outside or far from the center is minuscule(i.e., we assume a bell-shaped normal curve for the path of stock returns).

Figure 9.26 Stock price paths for constant �. Paths are bound not to jump discontinuously.


8 See R. Roll, ‘‘An analytic formula for unprotected American call options on stocks with known dividends’’, Journal ofFinancial Economics, 5, 251–258, 1977; R. Geske, ‘‘A note on an analytic valuation formula for unprotected American calloptions on stocks with known dividends’’, Journal of Financial Economics, 7, 375–380, 1979; and R. Whaley, ‘‘On thevaluation of American call options on stocks with known dividends’’, Journal of Financial Economics, 9, 207–211, 1981.

In the binomial setup, volatility translates directly to the width of the span of potentialoutcomes. Thus a 30% annual volatility may translate into a 70–20 span as in Example2b and a 40% volatility may result in a wider span of 80–15 as in Example 2c.If a dealer underestimates the actual volatility the stock will experience during the life

of the option, then he will discover that he has sold the option too cheaply. Thepremium he has charged will not cover the cost of the replicating strategy. In theBlack–Scholes model, that means the dealer should have used a higher volatilityinput. In the binomial model, the assumed span should have been wider. Note thatthe dealer is not asked to predict the really unpredictable (i.e., whether the stock will goup or down), but only the slightly unpredictable (i.e., whether the stock will move littleor a lot between now and expiry).Let us see in our binomial model what happens if the seller of a 55 call underestimates

volatility. Suppose he uses the model as in Example 2b, describing potential outcomesas 70 and 20. He charges $9 6

11for the call and borrows $5 5

11. With the total amount of

$15 he buys the prescribed 0.3000 shares at $50 a share. But the stock proves morevolatile (i.e., it attains either 80 or 15). The dealer will lose money whether the stockgoes up or down. Here is the diagram.

Figure 9.27 Example 2b revisited: mis-hedge.

If the stock goes up to 80, the 0.3 shares are worth $24, but the dealer owes $25 tothe option holder and $6 on his borrowing. If the stock goes down to $15, the 0.3shares are worth $4.50, but the dealer owes $0 to the option holder and $6 on hisborrowing.The primary risk of any option dealer is not the direction the underlying asset may

take, but the exposure to volatility of the underlying asset. A sold option results in ashort-volatility position. If the actual volatility of the underlying asset increases, thedealer loses money; if it decreases, the dealer makes money. A bought option resultsin a long-volatility position. If the actual volatility of the underlying asset increases, thedealer makes money; if it decreases, the dealer loses money. The sensitivity of an optionto the volatility input is called the vega of the option. The unit is the dollar change in thevalue of the option per 1% change in volatility. The vega of an option depends on thematurity of the option, the strike level relative to the current underlying asset’s price (in-the-moneyness), and the interest rate. In general, the longer the time to expiry the higherthe vega, as there is more time to lose money on mis-hedging. Also, the closer theoption is to the money the greater the vega. Deep in-the-money options and deep out-of-the-money options have low vegas.Option portfolios can be described as long or short volatility, too. A long-volatility

portfolio may contain bought and sold options, but the majority are bought. The


‘‘majority’’ here means that the net vega position in terms of net dollar sensitivity of theportfolio to a 1% change in the volatility (long-vega options minus short-vega options)is positive. A short-volatility portfolio has a net negative vega. With many bought andsold options in the portfolio, the overall vega of the portfolio changes over time and asthe price of the underlying asset fluctuates.

The vega of an option is related to another ‘‘Greek’’: the gamma. The latter relates tothe change in the delta of the option per unit of underlying price change. The higher thegamma, the more the delta hedge needs to be adjusted as a result of an underlying pricechange. High-gamma options are considered risky given that the required change in thedelta hedge may be difficult to execute quickly and without a loss.9 The vega (and thegamma) risk of an option portfolio cannot be hedged with positions in the underlyingasset, but only through option positions. A position in a stock does not have any vegaor gamma exposure; hence it cannot offset any option exposure. Only another optioncan.

Implied volatility

Running an option book is a bit of a chicken-and-egg game. In order to price theoptions we buy and sell, we need the volatility input into a pricing model (Black–Scholes, binomial, other). Different volatility assumptions will result in differenthedge ratios (i.e., positions in the underlying assets). Suppose we have somehowguessed the right input, priced all the options, computed the hedges, and bought orshorted the right net number of shares, bonds, or currency underlying the options inour portfolio. Our portfolio is now free of directional risk. Whether the underlying assetprice goes up or down, we do not show any profit or loss.

But how do we guess the volatility input? We could perform a statistical analysis ofthe past movements in the asset’s price and compute the standard deviation of thereturns on the asset. This historical volatility could then be input into the model.However, that would be tantamount to betting that the historical level of price varia-tion will continue into the future. A better way is to try to get at the market’s currentconsensus of the future price volatility. Where can we obtain that? In the option pricesother people charge.

Before we use the model to price our portfolio, we examine currently quoted optionprices. Dealers who quote these prices include their estimates of future volatility asinput into the same models to calculate their manufacture costs. We can back out whatthose estimates are by using our model to see what volatility input yields the prices thatthey quote. This implied volatility can then be input into our own model to price andhedge our portfolio.

Using implied volatility is always superior to using historical volatility as an inputinto an option model. This is because we can actually trade the options priced usingimplied volatilities. After all, we obtain those implied volatilities from actual optionquotes. We can use those options to eliminate our vega risk. Suppose our portfolio islong-volatility and market-neutral. That is, we have bought more options than we sold,but we bought or shorted enough of the underlying asset to eliminate the directionalrisk. We compute the sensitivity of our portfolio to a 1% change in the volatility input


9 Gamma of options is analogous to convexity of bonds. Both can lead to considerable mis-hedging.

and sell new options at the quoted prices by choosing the amount sold so that thoseoptions have the same sensitivity to a 1% volatility change. We also neutralize thedirectional risk of the newly sold options. Now our combined portfolio is directionalrisk-free and vega risk-free!The only way for a hedger to make money from trading options is to charge/pay for

options more/less than the fair value. That is, the hedger’s profit comes purely from thebid–ask spread and not from speculation on any explicit (price, rate) or implicit(volatility) market variable. Most dealers do not offset their vega risk completely andthus can be considered arbitrageurs in first-order risk (directional) and speculators insecond-order risk (volatility). They can offset their volatility risk but choose not to asthat would eat into their profits.The only time we may consider using historical volatilities as inputs is when there are

no options being quoted for expiries similar to the options in our portfolio. This istypical for very long-term options.

Volatility smiles and skews

In the Black–Scholes model, there is only one volatility input. This means that volatilityis constant over time. This is actually not a big problem and can be easily rectified. Wecan assume that volatility changes over time. We can then substitute a vector ofvolatilities, indexed by time, for a single number. In the binomial setup, this translatesinto considering different widths for the tree branches as we move through time steps(left to right). A bigger problem is associated with the fact that in the Black–Scholesmodel we also assume that volatility is constant across different price levels (up anddown in the tree). We can fix that by considering different branch widths across states.After all the ‘‘fixes’’, using perhaps a whole matrix of volatility inputs, the tree will mostlikely fall apart as different nodes will not recombine and the tree will more likely looklike our graph made up of separate price paths. This can still be handled by morecomplicated numerical methods. But what if, in reality, prices jump discretely and bymore than the normal distribution for return would allow, and what if they jumpdifferently at different levels and different times? What if the volatility itself is random?An option valuation model is never perfect. It is a simplification of potential price or

rate movements. The total scope of those movements is constrained by probabilisticassumptions about price paths and volatility structure. Paths close to the mean may beassumed more likely than those away from the mean. Paths may be assumed contin-uous (i.e., prices do not jump discretely). Volatility may be assumed constant in time,returns, or prices and unrelated to the price levels. There may be other unrealistic modelfeatures. In stock and currency models, the stock price or the FX rate may be assumedto fluctuate, but interest rates may not be. In interest rate option models, all rates maybe assumed to move in parallel or close to parallel. These simplifications are introducedto make the math of the models tractable.For these and perhaps many other reasons, it is well known that options on the

same underlying asset may require different volatility inputs for different strikes andexpiries.


First, consider the implied volatilities as published on October 17, 2003 by theFederal Reserve for at-the-money currency options as of the end of the previousmonth.

Table 9.1 Implied volatility rates for foreign currency options* (September 30, 2003)

1 week 1 Month 2 Month 3 Month 6 Month 12 Month 2 Year 3 Year

EUR 12.6 11.8 11.6 11.4 11.4 11.3 11.2 11.2

JPY 14.3 12.5 11.6 10.8 10.3 10.1 10.0 9.9

CHF 12.7 11.9 11.7 11.5 11.5 11.5 11.5 11.4

GBP 10.3 9.8 9.6 9.4 9.3 9.3 9.4 9.4

CAD 10.1 9.7 9.5 9.4 9.1 8.9 8.9 8.8

AUD 13.2 12.3 11.9 11.4 11.1 11.0 10.9 10.8

GBPEUR 8.3 7.8 7.7 7.6 7.6 7.5 7.5 7.3

EURJPY 12.8 11.6 11.1 10.7 10.5 10.3 11.6 10.3

*This release provides survey ranges of implied volatility mid-rates for at-the-money options as of 11:00 a.m.The quotes are for contracts of at least $10 million with a prime counterparty. This information is based ondata collected by the Federal Reserve Bank of New York from a sample of market participants and isintended only for informational purposes. The data were obtained from sources believed to be reliable but thisbank does not guarantee their accuracy, completeness, or correctness. For background information on therelease, see page VBGROUND.FRB http://www.ny.frb.org/markets implied.txtReproduced with permission of Federal Reserve Bank of New York (http://www.newyorkfed.org/markets/impliedvolatility.html)

For each currency, the implied volatilities change with the expiry date. Generally, theydecrease as the time to expiry increases. This implies that dealers estimate differentreplication costs for short options than for long options. This phenomenon perhapshas to do with the distinction between the realized and implied volatility and meanreversion. Short-term at-the-money options will require constant rebalancing as thehedge ratio computed between now and expiry will fluctuate. For longer termoptions, the hedge ratio day-to-day is going to change much less (longer time toexpiry may make the present value of payoff less variable); therefore, while theactual volatility of the FX rate per day may be the same over a short period as overthe long period, the dealer’s ‘‘realized’’ volatility will be lower. The dealer will performthe buy-high-sell-low unprofitable trades less frequently. Therefore, his cost of manu-facture or the premium will be lower. This will result in a lower computed impliedvolatility. Mean reversion refers to the possibility that, while FX rates fluctuate un-predictably in the short run, they tend to oscillate around long-term trend lines and thefurther they deviate from the trend lines the more they are pulled toward them (i.e.,volatility depends on the level of rates). As option models cannot take all of thesepossibilities into account, they can be ‘‘fixed’’ by lowering the implied volatility forlong-term options to reflect the lower cost of manufacture.

Consider another example of the computation of implied volatilities, as of October15, 2003; this time for options on the S&P 500 index futures expiring December 18,2003.


Table 9.2 December futures on the S&P 500

Strike Implied volatility Implied delta Vega——————————— ——————————— (ticks)Call Put Call Put

975 20.81 20.49 0.79 �0.2 1.24980 20.48 20.21 0.78 �0.21 1.29985 20.18 19.95 0.76 �0.23 1.34990 19.91 19.72 0.75 �0.25 1.39995 19.59 19.43 0.73 �0.26 1.441000 19.36 19.16 0.71 �0.28 1.491005 19.07 18.91 0.69 �0.3 1.531010 18.79 18.66 0.67 �0.32 1.571015 18.52 18.42 0.65 �0.34 1.611020 18.19 18.12 0.63 �0.36 1.651025 17.92 17.87 0.61 �0.38 1.681030 17.76 17.68 0.58 �0.41 1.711035 17.53 17.48 0.56 �0.43 1.731040 17.29 17.27 0.53 �0.46 1.741045* 17.04 17.04 0.51 �0.48 1.751050 16.88 16.91 0.48 �0.51 1.751055 16.65 — 0.45 �0.54 1.741060 16.46 16.54 0.43 �0.56 1.731065 16.3 — 0.4 �0.59 1.71070 16.12 16.19 0.37 �0.62 1.671075 15.91 16.02 0.34 �0.65 1.621080 15.81 — 0.32 �0.67 1.571085 15.75 — 0.29 �0.70 1.521090 15.62 — 0.27 �0.72 1.451095 15.53 — 0.24 �0.75 1.391100 15.43 15.64 0.22 �0.77 1.32

December futures ¼ 1044.50, days ¼ 47, at-the-money volatilities ¼ 17.06%, interest rate ¼ 6.50%. Source:http://www.pmpublishing.com/volatility/sp.html#StandardDeviations

Figure 9.28 S&P 500 implied volatility skew (December 18, 2003). Puts with strikes below 1045;calls with strikes above 1045.Source: http://www.pmpublishing.com/volatility/sp.html#StandardDeviations.



The at-the-money level is 1044.50 (¼ current value of the index). Calls/Puts with strikeshigher/lower than that level are called out-of-the-money, and with strikes lower/higherthan that level they are called in-the-money. The table shows the implied volatilities fordifferent strike levels. As a general rule they decrease as the strike level increases in ahalf-smile fashion as shown on the enclosed graph (for very high strikes they startincreasing again slightly). This relationship of implied volatilities to strikes is generallyreferred to as a volatility smile or volatility skew. It is a reflection of all the simplificationswe cited above. It can, for example, be explained by the fact that as prices drop theytend to drop and fluctuate by more than assumed by the standard model (volatility ishigher at lower price levels), and as prices increase they tend to increase more graduallywithout big jumps (volatility is lower as prices increase). The existence of volatility skewintroduces additional risk to option portfolios. Options on the same underlying andwith the same maturities, but with different strike prices, are not good hedges for eachother. Hedging with imperfect substitutes carries basis risk (similar in a way to that ofhedging one commodity with another, e.g. jet fuel with oil). Explicit modelling of thevolatility skew allows dealers to minimize the tertiary risk to relative volatility changesacross strikes. They involve relating different, implied volatilities through a postulatedmathematical function, itself often assumed not to change over time.

Table 9.1 also shows the delta for each option sold, in units of futures contracts to bebought/shorted per one option, as well as its vega, in ticks (index points) per 1%volatility change.

9.9 INTEREST RATE OPTIONS, CAPS, AND FLOORS

The modeling of options on interest rates is more complicated than those on equityprices, commodity prices, or FX rates. There are several reasons for that. First andforemost is that interest rates are not prices of assets that can be bought and sold.Imagine in our binomial example that, instead of stock prices, we place the rate on a10-year bond on the nodes of the tree. We build the tree using the right volatility andfollow our recipe to a tee. Now we try to hedge using the recipe’s prescriptions. We wantto dynamically trade the underlying asset so that the dollar changes in our hedgeposition exactly offset the changes in the value of the option over its entire life. In thestock example, the delta of the option told us how many shares of the asset to buy orsell. In the commodity case, that would be the amount of the commodity; in thecurrency case, that would be the amount of foreign or domestic currency. As theprice of the stock, commodity, or currency changed by one dollar, the value of ourhedge position changed exactly by delta times one dollar. The complication with interestrates is that we cannot buy an interest rate. We can buy the 10-year bond whose price(PV of cash flows) depends on the rate, but not the rate itself. Suppose we try tooutsmart the option model and we buy or sell some amount of the bond correspondingto the ‘‘amount’’ of the rate we were supposed to buy or sell as our delta. The hedge willnot work because the relationship between bond price changes and rate changes is non-linear (duration is not constant in yield). As the rate changes, what seemed like the rightamount of the bond to buy will prove to be slightly wrong, but will not be equallywrong, depending on whether the yield goes up or down. The dollar value change in thehedge will not be equal to the delta times 1 bp change in the rate. This convexity featureof the interest rate instruments can be partly overcome by the use of non-convex futuresinstruments. A Eurocurrency contract is the best example. Its value (variation margin


settlement) is a constant monetary amount per 1 bp change in the interest rate change.But the application of non-convex instruments can only help in a limited number ofcases. Most spot or forward rates on which options are written will be themselvescomplicated functions of the intermediate futures. The 10-year bond as an asset canbe synthesized from shorter spot rates and lots of intermediate forwards. The 10-yearpar rate will still be a very non-linear function of those rates or forward prices.The second important complication for interest rate options is that interest rates are

not independent of each other. If the 10-year par rate can be synthesized from 40quarterly forward rates, then do we have to take into account the volatilities of allthose rates and their correlations? Our binomial nodes then have 40 variables on them.And, then, how do we come up with all the volatility and correlation inputs? It was hardenough with a single stock, let alone with tens of interrelated rates.The solution is often a reduced-form model with only one or two variables driving

the uncertainty in the model. All other rates are then assumed to be deterministicfunctions of that one state variable, whose implied volatility structure we are confidentabout (i.e., we can calibrate from other existing options). If the model has two statevariables, then we must be able to observe enough prices for other options so that wecan not only back out implied volatilities for both state variables, but also the correla-tions between the two (in a process analogous to the implied volatility computations).The assumption of one or only two state variables is clearly unrealistic, but most of thetime this is the best we can do. Often, this is accurate enough. But because the mathbehind models like these becomes quite complicated, only experts can tell whether weare close enough to the true value of the option and the true hedge ratios.

Options on bond prices

Suppose we buy a call option on a price, not a rate, of a specific 5-year bond struck at102. Can we price the option by assuming some volatility of the bond price andconstruct our binomial tree? Unless the option expiry is super-short, the answer isstill probably no.Bond prices do not fluctuate like stock prices. First, each bond has a maximum price

equal to the sum of the coupons and principal (when the yield is 0, the present valueboils down to a simple sum as all discount factors are equal to 1). Second, the bondprice must be equal to the face value by maturity. In a way, the bond price is pulledtoward that known final value. This constrains how much the bond price can fluctuate.A bond with little time to maturity cannot move much, even if the yield changesdramatically. Stocks or commodity stock prices do not have maturities and hencepredetermined maturity prices.Hence even options written on bond prices, not rates, may require a rate-driven

valuation model with attendant complications as described above.

Caps and floors

Caps and floors are packages of options, calls, and puts, correspondingly, on the sameunderlying (mostly short-term) interest rate with sequential expiry dates. The vastmajority of caps and floors have the 3-month LIBOR as their underlying rate. Theexpiry dates on the options, called caplets or floorlets, follow a quarterly or semi-annualschedule, typically matched to the maturity of the underlying rate, starting with im-mediate expiry and ending with the last expiry date being one period prior to the statedmaturity of the cap. For example, a 5-year cap on 3-month LIBOR struck at 4.5%

consists of 20 caplets with expiries of 0 months, 3 months, 6 months, 9 months, and soon, all the way to 57 months. The payouts on the options are delayed by 3 months (i.e.,made in arrears); the last payout is on the maturity date of the cap (i.e., in 60 months).The payoff on each caplet is equal to the greater of 0 and the difference between theunderlying LIBOR rate on the expiry date and the strike rate of 4.5% times the day-count fraction on an Act/360 basis for the period covered by the LIBOR rate. A floor isa ‘‘put’’ equivalent of a cap. That is, the payoff of each floorlet is equal to the greater of0 and the difference between the strike rate of 4.5% and the underlying LIBOR rate onthe expiry date times the same day-count fraction. All these details are designed tomatch the swap market conventions, since caps and floors are viewed as natural supple-ments to swaps. Let us illustrate our 4.5% cap and 4.5% floor by assuming somehypothetical LIBOR rates (in column 2) for future option expiry dates (called setdates), and day counts for the subsequent 3-month period until the pay dates. Theprincipal amount is $100,000,000.

Table 9.3 Five-year, 4.5%, $100 million cap and floor on 3-month LIBOR (dates in monthsfrom today)

Set date LIBOR Cap Floor Pay date——————————————— ———————————————Max(L� K , 0) Days Payout Max(K � L, 0) Days Payout

0 4.50 0 91 0 0 91 0 33 4.20 0 91 0 0.3 91 75,833 66 4.81 0.31 92 79,222 0 92 0 99 5.20 0.7 90 175,000 0 90 0 1212 5.40 0.9 89 222.500 0 89 0 1515 5.55 1.05 91 265,417 0 91 0 1818 5.83 1.33 92 339,889 0 92 0 2121 6.21 1.71 91 432,250 0 91 0 2424 6.43 1.93 90 482,500 0 90 0 2727 6.11 1.61 92 411,444 0 92 0 3030 5.73 1.23 91 310,917 0 91 0 3333 5.32 0.82 91 207,278 0 91 0 3636 5.17 0.67 89 165,639 0 89 0 3939 4.85 0.35 92 89,444 0 92 0 4242 4.62 0.12 91 30,333 0 91 0 4545 4.33 0 91 0 0.17 91 42,972 4848 4.02 0 89 0 0.48 89 118,667 5151 3.78 0 92 0 0.72 92 184,000 5454 3.66 0 91 0 0.84 91 212,333 5757 3.21 0 90 0 1.29 90 322,500 60

The buyer of the cap benefits if spot LIBOR rates on future dates exceed 4.5%. Thebuyer of the floor benefits if spot LIBOR rates on future dates are below 4.5%. Typic-ally, the first caplet and floorlet are deleted, unless specifically stated to the contrary(i.e., technically we have 19 optionlets). Caps can be viewed as protection againstinterest rate increases, while floors can be viewed as protection against interest ratedeclines.

The at-, in-, and out-of-the money terminology for caps and floors is a little differentfrom stock options; there, we usually had only one option, here we have many. A cap orfloor is said to be (struck) at-the-money if the strike price is chosen to be the swap rateof the same maturity as the maturity of the cap. Correspondingly, we can talk about


in-the-money or out-of-the-money caps and floors, depending on whether their strike isgreater or less than the swap rate. The statement applies on an aggregate basis, not toindividual optionlets. In our illustration, if the 5-year swap rate on an Act/360 basis is5%, then our cap would be called in-the-money (as the swap rate is higher than thestrike), even though based on today’s LIBOR the caplets are really at-the-money and,on a forward basis, some caplets may be out-of-the-money. Our floor would be out-of-the-money as the swap rate is above the strike.

Relationship to FRAs and swaps

We have shown before that a swap is a package of forward-rate agreements (FRAs). A5-year quarterly swap can be viewed as a package of 20 FRAs with subsequent matu-rities. The start date of each FRA matches one set date of the swap, and the end date ofthe FRA matches the pay date of the swap corresponding to the given set date. The firstFRA is 0� 3, the next 3� 6, and so on. The last one is 57� 60.Instead of dissecting swaps along set and pay dates, we can dissect them along rate

levels. A pay-fixed swap can be viewed as a long cap and a short floor. Similarly, eachconstituent FRA can be viewed as a long-caplet and short-floorlet position. Suppose webuy a $100 million 5-year cap struck at 4.5% and sell a $100 million 5-year floor struckat 4.5%. When LIBOR exceeds the strike, the payoff on the floor is 0 and the payoff onthe cap is equal to LIBOR minus the fixed rate of 4.5%. This is equivalent to a receiptof LIBOR and payment of fixed 4.5%. When LIBOR is below the strike, the payoff onthe cap is 0 and the liability on the floor is equal to the fixed rate of 4.5% minusLIBOR. This is equivalent to a receipt of LIBOR and payment of fixed 4.5% (i.e.,no matter where LIBOR sets, the cash flows are equivalent to those of a swap with afixed rate of 4.5%). Table 9.4 summarizes the situation.

Table 9.4 Five-year, 4.5%, $100 million cap, floor, and swap (dates in months from today)

Set date LIBOR Days Long cap Short floor Pay date Swap—————————————— ——————————————— ————————————————————

Max(L� K, 0) Payout Max(K � L, 0) Payout Receive Pay Net

0 4.50 91 0 0 0 0 3 1,137,500 1,137,500 03 4.20 91 0 0 0.3 �75,833 6 1,061,667 1,137,500 �75,8336 4.81 92 0.31 79,222 0 0 9 1,299,222 1,150,000 79,2229 5.20 90 0.7 175,000 0 0 12 1,300,000 1,125,000 175,00012 5.40 89 0.9 222,500 0 0 15 1,335,000 1,112,500 222,50015 5.55 91 1.05 265,417 0 0 18 1,402,917 1,137,500 265,41718 5.83 92 1.33 339,889 0 0 21 1,489,889 1,150,000 339,88921 6.21 91 1.71 432,250 0 0 24 1,569,750 1,137,500 432,25024 6.43 90 1.93 482,500 0 0 27 1,607,500 1,125,000 482,50027 6.11 92 1.61 411,444 0 0 30 1,561,444 1,150,000 411,44430 5.73 91 1.23 310,917 0 0 33 1,448,417 1,137,500 310,91733 5.32 91 0.82 207,278 0 0 36 1,344,778 1,137,500 207,27836 5.17 89 0.67 165,639 0 0 39 1,278,139 1,112,500 165,63939 4.85 92 0.35 89,444 0 0 42 1,239,444 1,150,000 89,44442 4.62 91 0.12 30,333 0 0 45 1,167,833 1,137,500 30,33345 4.33 91 0 0 0.17 �42,972 48 1,094,528 1,137,500 �42,97248 4.02 89 0 0 0.48 �118,667 51 993,833 1,112,500 �118,66751 3.78 92 0 0 0.72 �184,000 54 966,000 1,150,000 �184,00054 3.66 91 0 0 0.84 �212,333 57 925,167 1,137,500 �212,33357 3.21 90 0 0 1.29 �322,500 60 802,500 1,125,000 �322,500


Each row is a dissection of a FRA into a cap and a floor. The sum of the rows is equalto the swap. Each column group is a dissection of the swap into a long cap and shortfloor. The net receipt on the swap is identical to the net of the cap and floor positions:

Cap struck at K � Floor struck at K ¼ Swap with fixed rate K

Suppose that we do not pick the strike rate randomly. Instead we select it in such a waythat the premium we pay on the cap is equal to the premium we receive on the floor (i.e.,we have no net cash flow upfront). What swap would have no net cash flow upfront? Apar swap or an on-market swap. We can thus conclude the following:

Capðwith K ¼ Swap rateÞ � Floorðwith K ¼ Swap rateÞ ¼ Par swap

Recall the analogous relationship for stock options struck at the forward. We showedthat as implied volatilities increased the values of the calls and puts, both struck at theforward, increased by the same amount as their payoff was still equivalent to theforward. The same is true for caps and floors. In order to conform to the constraintof arbitrage, if prices of caps struck at the swap rate increase (due to implied volatilityrise), the prices of floors struck at the swap rate must also increase by the same amount.The pricing assumptions do not change the payout on the combined position, which isstill equivalent to an on-market swap. These considerations are the basis for the in-the-moneyness language for caps and floors as defined above.

We can construct swaps out of caps and floors and vice versa. A cap can be viewed asa combination of a pay-fixed swap and a long floor position. A floor can be viewed as acombination of a long cap position and a receive-fixed swap.

An application

Bond issuers often combine a long cap position with a floating-rate bond. This ensuresthat the coupon payments on the bond do not exceed a certain desired level.

Figure 9.29 An issuer of a floating rate bond caps interest liability of 4.5%.

Suppose a company issues a 5-year bond paying quarterly floating coupons equal toLIBOR. The company also buys from a dealer a cap struck at 4.5%. If LIBOR on anycoupon set date exceeds 4.5%, the company’s net obligation will stay at 4.5%. IfLIBOR on any coupon date is below 4.5%, the company takes advantage of thefloating nature of the issue and pays less than 4.5%. Its net obligation is thus the


lower of the two: LIBOR or 4.5%. The hypothetical cash flows are summarized inTable 9.5.

Table 9.5 Five-year, $100 million floating rate bond and 4.5% cap (dates in months from today)

Set date LIBOR Days Long cap Floating bond Pay date Net——————————— couponMax(L� K, 0) Payout

0 4.50 91 0 0 �1,137,500 3 �1,137,5003 4.20 91 0 0 �1,061,667 6 �1,061,6676 4.81 92 0.31 79,222 �1,229,222 9 �1,150,0009 5.20 90 0.7 175,000 �1,300,000 12 �1,125,00012 5.40 89 0.9 222,500 �1,335,000 15 �1,112,50015 5.55 91 1.05 265,417 �1,402,917 18 �1,137.50018 5.83 92 1.33 339,889 �1,489,889 21 �1,150,00021 6.21 91 1.71 432,250 �1,569,750 24 �1,137,50024 6.43 90 1.93 482,500 �1,607,500 27 �1,125,00027 6.11 92 1.61 411,444 �1,561,444 30 �1,150,00030 5.73 91 1.23 310,917 �1,448,417 33 �1,137,50033 5.32 91 0.82 207,278 �1,344,778 36 �1,137,50036 5.17 89 0.67 165,639 �1,278,139 39 �1,112,50039 4.85 92 0.35 89,444 �1,239,444 42 �1,150,00042 4.62 91 0.12 30,333 �1,167,833 45 �1,137,50045 4.33 91 0 0 �1,094,528 48 �1,094,52848 4.02 89 0 0 �993,833 51 �993,83351 3.78 92 0 0 �966,000 54 �966,00054 3.66 91 0 0 �925,167 57 �925,16757 3.21 90 0 0 �802,500 60 �802,500

For months 9–45, when LIBOR exceeds 4.5%, the company effectively pays a 4.5%rate times the appropriate day-count.A mirror image application of a long floor position is on the investment side.

Suppose a portfolio manager owns a floating rate bond and fears that as rates comedown her income from the bond will decline. She can purchase a floor struck at thedesired level to maintain her income at or above that level. In the discussion of swaps,we also showed additional uses of caps and floors in structured finance.

9.10 SWAPTIONS

Swaptions are options to enter into a swap. Unlike a cap or floor with a series of expirydates for all optionlets, a swaption has one expiry date. Once the holder exercises aswaption, he will pay and receive multiple cash flows, like on a swap. Exercise can beEuropean-style (once at expiry only), American-style (once any time prior to or onexpiry date), or Bermudan-style (once on any swap date prior to or on the expiry date).For example, a 3-into-7, $10 million, European call swaption struck at 5% gives theowner the right to receive a fixed 5%, against floating LIBOR, on a $100 million 7-yearswap. If the option is exercised on the expiry date, which is 3 years from today, the swapwould start on that day and end 7 years later. The same swaption can be referred to as


3-year-10-year-final swaption to imply that the exercise right is in 3 years and the finalmaturity of the swap is in 10 years. A Bermudan or American version would beexercisable between today and 3 years from today, not just 3 years from today, andthe swap would start immediately at exercise and end 10 years from today. The ‘‘3-into-7’’ language is rarely used with American and Bermudan options; the ‘‘3-year-10-year-final’’ language is preferable (if the option is exercised in 2 years, the swap will last 8years).

Calls are options to receive (fixed) on the swap; puts are options to pay (fixed) on theswap. The call/put terminology corresponds to the view of swaps as exchanges ofbonds. A receive-fixed swap can be thought of as a bought fixed-rate bond and asold floating-rate bond, or a bought fixed-rate bond financed by a revolving loan.So, a call swaption is like an option to buy a fixed-rate bond, just like a call is anoption to buy a stock. A put swaption is an option to sell a fixed-rate bond (i.e., theoption to pay a fixed rate on an obligation, and to receive a floating financing rate).

Options to cancel

Swaptions can be packaged with swaps to provide options to cancel the swap. Supposewe pay a fixed rate of 4.5% on a 10-year quarterly swap and we receive 3-monthLIBOR. Suppose also our swap counterparty sells us a 5-into-5 call swaption struckat 4.5. The call gives us the right to receive 4.5% on a 5-year swap starting 5 years fromtoday. But if we exercise the call, then we will exactly offset the remaining cash flows onthe existing 10-year swap.

Figure 9.30 A 5-into-5 call swaption packaged with a 10-year swap.

The call swaption, which we defined as the right to enter a swap to receive fixed, canalso be defined as the right to cancel a pay-fixed swap. The ‘‘call’’ language thusconforms to the call provisions on fixed coupon bonds. Analogously, the ‘‘put’’notion for swaptions corresponds to the right of the fixed coupon bond holders toput the bonds back to the issuer at par.

Relationship to forward swaps

From the above construction, we should also be able to see the following relationship:

Call swaption� Put swaption ¼ Forward swap


where Forward swap is defined as a swap with the first set date in the future (and nottoday).Suppose we buy a 5-into-5 call struck at 4.5% and sell a 5-into-5 put struck at 4.5%.

If 5-year swap rates are low in 5 years, say at 3%, then we will exercise our call right toreceive fixed 4.5% (above market). The holder of the put we sold will not exercise.Alternatively, if 5-year swap rates are high in 5 years, say at 6%, then the holder of theput we sold will exercise his right to pay fixed 4.5% (below market). We will not exerciseour call. His exercise decision will force us to receive fixed 4.5%. Thus no matter whatswap rates are in 5 years, we will enter into a 5-year swap at that time. Viewed fromtoday, this is a forward swap. It starts at a future known date and ends at a futureknown date, and the fixed rate on it is agreed on today.In addition, if instead of 4.5% we choose the strike rate in such a way that the

premium paid for the call equals that received for the put, then the sure forwardswap will be arranged at no cost to either party (i.e., it will be a par forward swap).This relationship will be true no matter what the level of implied volatilities used by

dealers as inputs into their pricing models, as the static arbitrage constraint will notchange.Swaptions can be synthesized from forward swaps and other swaptions. A call is

equivalent to a forward receive-fixed swap and a put swaption. A put is equivalent to aforward pay-fixed swap and a call swaption.Swaptions can also be viewed as one-time options on long (swap) rates. Note that, in

the above discussion, we decided that the call holder will exercise when future swaprates are lower than the strike. This guarantees that the present value of the swap hechooses to exercise into is positive (i.e., he has a positive payoff). This is true becausethe LIBOR leg part of the swap (equivalent to the floating-rate bond) always prices(present-values) to par, and the positive PV will come from the discounted value of thedifferences between the strike and the actual lower fair swap rate (i.e., the bond with afixed coupon equal to the strike will price above par). If the fair rate were equal to thestrike, the fixed bond with a coupon equal to the strike would also price to par. So thepayoff of the call can be viewed as the difference between the strike and the fair swaprate times the day-count-corrected annuity factor (sum of discount factors for all swappayment dates):

Call ¼ Max½0;K � Swap rate� � Annuity factor

Put ¼ Max½0;Swap rate� K � � Annuity factor

Swaptions and caps and floors are also related, but not so simply. Both types of optionsare ways of dissecting swaps (i.e., share the risk of the swap with other players). Capsand floors can be forward starting to make them look identical to swaptions in terms ofstart and end dates. But caps and floors dissect swaps on the floating side, whileswaptions do so on the fixed side. Caps and floors are packages of several mini-options on each swaplet (FRA), while swaptions are one-time options on the entireswap. The two are related for the following reason: the long (forward) swap rate, whichis the underlying rate for the swaption, is a package of forward-starting swaplets(FRAs). That one rate can be exchanged costlessly into a series of short-term fixedrates (equal to FRA rates for the respective pay periods) or further into unknown


floating-rate payments (as on-market FRAs cost nothing to enter into). A long cap andshort floor position is equivalent to those floating-rate payments.

Because of these interrelationships between long and short rates, we can claim thatthe prices of swaptions and caps and floors are interrelated, as are their impliedvolatilities. The modeling of these relationships is extremely hard.

9.11 EXOTIC OPTIONS

The term ‘‘exotic options’’ applies to all non-standard options (i.e., with payoffs notdefined like those for calls and puts) traded OTC. Exotic options include digital, orbinary, options (fixed point payoffs irrespective of how far into the money the optionis), barrier options, like knock-ins or knock-outs (the price or rate has to hit or avoidcertain barriers prior to expiry for the payoff formula to even apply), as well as a wholevariety of options that are difficult to price because the primary risk in them is not easilyhedgeable. We will not attempt a complete listing of the exotics. Instead, we focus onsome very popular structures that at first appear to be quite simple. But, they areanything but simple.

Periodic caps

Consider a fairly standard provision of an adjustable mortgage. The interest ratechanges once a year based on some floating index, but the rate is guaranteed not tochange by more than 2% per year or 6% total over the lifetime. The mortgagee has ineffect issued a floating-rate bond to finance a house purchase. Every year, the interestrate he pays is based on the cost of funds for that year (set at the beginning of the yearbased on some 1-year money rate). Next year the rate adjusts up or down. But the rateis guaranteed not to go up by more than 2% year to year. So if the index changes from3% to 6%, the homeowner’s rate increases only by 2% instead of 3%. The option thehomeowner holds is not a standard call option on a rate or a cap. The strike onthe option changes every year and is based on last year’s rate and last year’s strike.If the following year the index changes from 6% to 9%, the mortgage rate again onlygoes up by 2% and only from the already ‘‘unfairly’’ low last year’s level that did notreflect a full index increase at that time. Mortgage banks that want to protect them-selves against income lost due to these imbedded options purchase periodic caps fromdealers. Periodic caps pay the difference between the mortgage rate (with the options)and the fully indexed floating rate (without the options). The resetting strike featurechanges the probability of payoff relative to a straight call. It also changes how thepayoff can be manufactured. The dealer-hedger cannot compute the delta on the optionuntil he knows the strike. Thus it is not the implied volatility over the entire life of theoption that determines the cost of manufacturing, but it is the sequence of futureimplied volatilities on shorter 1-year options. This is true with the caveat that previousstrikes also carry over as in our example. The cost of payoff manufacturing has thus aknown component (stickiness of the strike) and some unknown component (futureannual-rate differences). The premium quoted will reflect the subjective bet on thefuture path of implied volatilities.


Constant maturity options (CMT or CMS)

Constant maturity options are caps or floors with long-term government (CMT) or swaprates (CMS) as the underlying variables, but with the day-count and frequency ofpayoff like that of a short rate. A 5-year quarterly cap on a 10-year rate struck at6% pays the greater of 0 and the difference between the 10-year rate and the strikeevery quarter for the next 10 years. The day-count used for the payoff calculations isthat of a given quarter. The payoff is made in arrears, just as the case is for a simple cap.The only difference between a straight cap and a CMS cap is that, instead of comparingLIBOR to the strike, we compare the 10-year swap rate to the strike. Why does thispose problems?The simple answer is: convexity. With a standard cap, the hedge instrument is a 3-

month LIBOR deposit, a forward, or a futures on it. While these may be non-linearinstruments (i.e., the PV of the position changes as the rate changes, but not as a fixedmonetary amount per 1 bp rate change), the non-linearity is nowhere near as great asthat of longer term instruments and can be fairly easily corrected for. With CMS rates,as we compute the delta per 1 bp move and then translate it through duration into adollar holding of the underlying bond, our hedge will always be imperfect and differ-ently so on the up- and downsides. These hedging errors will magnify, the longer thehedge. On a 5-year maturity option, they can be very large. The CMT options have theadded difficulty of swap spread exposure.We can argue that these instruments are more similar to swaptions than to caps.

Recall that a swaption was a bet on a future long-term rate. The payoff was multipliedby the appropriate annuity factor, reflecting the final maturity of the underlying swap.Here, a CMS cap can be seen as a series of swaptions with increasing maturities andwith payoffs multiplied by an ‘‘inappropriate’’ annuity factor, one for a cap instead offor a swaption. Because that annuity factor is correct for swaptions, they can be hedgedeasily with their underlying instruments (i.e., forward swaps), which in turn can besynthesized from long long-maturity swaps and short short-maturity swaps. CMScaps produce a hedge error from the mismatch of the actual annuity factor with thatof the underlying forward swap.

Digitals and ranges

A digital option is an option with a fixed monetary payoff, if a price or rate breaches acertain strike level. A range is a compound version of a digital option where the under-lying price or rate has to breach one level but not go over or under another level (i.e., ithas to end up within a predetermined range). These structures are popular in commod-ity, FX, and interest rate markets. In interest rate markets, they are typically packagedinto cap- or floor-like serial forms with short-term LIBOR rates as the underlyingsand range levels changing period to period. These options are not difficult to pricetheoretically, but they pose a risk of low-probability events with highly uncertainhedge outcomes.Suppose we sold an FX range that pays $1 if the USD/EUR rate is within 1.10 and

1.20. We price the option using a binomial tree and have followed the hedge recipe. Wecome to 1 day prior to expiry. The spot USD/EUR FX rate stands at 1.20. If it ticks upby tomorrow, we will owe nothing to the option holder. If it ticks down we will owe $1


(everything). There is no effective hedge strategy that will produce the desired payoff: $0or $1. We have to gamble. At the time the option was sold, this knife-edge event hadeffectively a zero probability. In a standard call or put, these events do not occurbecause the size of the payoff changes monotonically with the level of the rate, allowingus to adjust the hedge. Here it jumps from 0 to everything over a practically non-existent move in the underlying.

Quantos

Quantos are options whose payoffs are defined in non-native currencies. A seeminglystandard put option on the FTSE 100, whose payoff is in constant U.S. dollars perpoint of the index, is quite a bit more complicated than that whose payoff is defined inBritish pounds. In the latter case, the hedge is obvious. The seller of the put shorts thestocks in the index. The total change in their value in pounds dynamically produces thepound payoff at expiry. The quanto version of the option forces the dealer into anadditional currency hedge as the pound payoff needs to be guaranteed in dollars. Inaddition, the two hedges are interrelated: as the potential payoff in dollars rises becauseof the FTSE change, the underlying stock hedges pound value may over- or under-compensate for a possible FX rate change when the hedge is liquidated and the payoffmade.

The quanto label can be added to almost any option. A popular equity option thatpays the best of several national indices (say, S&P 500, Nikkei 225, and FTSE 100) isoften quantoed into one desired currency, say the euro.


______________________________________________________________________________________________________________________________________________________________________ 10 ______________________________________________________________________________________________________________________________________________________________________

____________________________________________________________________________________________________________ Option Arbitrage ____________________________________________________________________________________________________________

We review relative value strategies with options. We start with static cash-and-carrytrades where the arbitrage principles are the same as with non-option products, but weexploit the relative mispricing of forwards as long call–short put positions. Thishappens frequently when one or more options needed to synthetically replicate aforward can be acquired very cheaply as parts of more complex securities. Next weconsider dynamic arbitrage strategies where the underlying market risk is hedged outand the trade is a relative value bet on secondary risk. The secondary risk involved canbe a relative play on two related, but not identical implied volatilities: correlation riskor spread. We end with dynamic strategies spanning several asset classes and someexotic examples with truly unhedgeable risks.To readers interested in options, our treatment of dynamic option strategies will most

likely seem wanting. We only sketch out some basic trades. Relative value arbitrage inoptions can get very complicated. The residual risks can be significant, unhedgeable,and their understanding may require math knowledge that goes beyond the scope ofthis book. They can be defined in terms of exposures to volatility skews, to correlations,to local vs. term volatilities (i.e., period to period vs. total end dispersion), often insecond- and third-order derivative terms. They can also involve tradeoffs betweenexposures from different markets, stock price correlation with interest rates, creditspread correlation with interest rates, or interest rates with currency rates. Our goalfor this chapter is limited to highlighting the risks of some benchmark trades.

10.1 CASH-AND-CARRY STATIC ARBITRAGE

To make clear the static arbitrage principle, we start with a simple financing trade:borrowing or lending against a known, non-contingent cash flow constructed by com-bining several contingent payoffs.

Borrowing against the box

The borrowing-against-the-box strategy can be employed with options on any under-lying: stocks, currencies, commodities, etc. The idea is very simple. If a long call–shortput position is equal to a long forward and a short call–long put position is equal to ashort forward then by combining four options with the same expiries into a syntheticlong forward at one price with a synthetic short forward at another, we can generate asure payoff at the expiry day. If the net premium (today) for the four options is positive(an inflow), then the payoff at expiry is an outflow and the strategy is equivalent to

borrowing. If the net premium is paid (outflow) and the payoff is an inflow, then thestrategy is equivalent to lending. The arbitrage consists of trying to lock in through thisstrategy a higher lending, or lower borrowing rate than our own cost of financing.

Let us consider a stock option case using the multi-step pricing model from Chapter9. We employ a three-step binomial to price four 3-month options with the inputs asbefore. Stock currently trades at 50; the interest rate is 2.6% per month (de-annualized). The forward stock price for delivery 3 months from today can be com-puted as 54.0023. We sell a 50 call and buy a 50 put (short synthetic forward at 50). Webuy a 55 call and sell a 55 put (long synthetic forward at 55). As each synthetic forwardis off-market, the net premium is non-zero. Table 10.1 summarizes our position.

Table 10.1 Borrowing against the box

Position Strike Option Premium

Short 55 Put 7.6497Long 55 Call �6.7259Long 50 Put �6.3419Short 50 Call 10.0475

————

Net 4.6294

Let us consider the payoff. If the stock at expiry is 45, the calls are worthless, we pay 10on the 55 put, and receive 5 on the 50 put; our net is an outflow of 5. If the stock is at72, the puts are worthless, we receive 17 on the 55 call, and pay 22 on the 50 call; our netis an outflow of 5. If the stock is at 52, we pay 3 on the 55 put, the 55 call and the 50 putare worthless, and we pay 2 on the 50 call; our net is an outflow of 5. No matter wherethe stock ends up at expiry, out net cash flow will be an outflow of 5. So, we traded a netinflow of 4.6294 today for a net outflow of 5 three months from today. What interestrate is implied in this synthetic borrowing of 4.6294? From:

4:6294ð1þ rÞ3 ¼ 5:0000

we compute the rate to be 2.6% per month. This is exactly what we assumed for valuingthe four options.

Suppose we have an investment opportunity to earn 3.2% per month on comparablerisk. That is, we know that if we invest:

5:0000=ð1þ 0:032Þ3 ¼ 4:549 157

the investment will accrue to exactly 5.0000. We synthetically borrow 4.6294 and invest4.5492; our net profit today is 0.0803. When the investment matures in 3 months, wecollect 5 and distribute it appropriately to the option claimants. The only risk we arefacing is that of default on our investment. If, for some reason, the investment fails toproduce 5, we will not be able to meet our obligations under the option contracts. Wehave statically eliminated any market risk of stock fluctuations by choosing a combina-tion of options with a known payoff. We do not have any market risk of interest ratefluctuations, because we are not going to hedge the options (i.e., we are not going tofinance any stock positions). We are simply left with default risk.


Index arbitrage with options

Recall from Chapter 7 that stock index arbitrage consisted of either a long index futuresposition against a short spot basket with lending or a short index futures positionagainst a long spot basket with borrowing to buy the basket. The direction of thetrade depended on the fair value calculation for the futures, and the spot basket plusmoney market transaction was equivalent to a synthetic forward. Here we are going toexploit the put–call parity of Chapter 9 to construct the forward. Since the long forwardposition is equivalent to a long call and short put position (both struck at the forward),we will be able to perform a static cash-and-carry by transacting in options upfront andtrading them against futures or basket positions.Suppose, as in the previous example, that the stock index trades at 50 and the interest

rate is 2.6% per month. The 3-month futures contract on the index trades at its theo-retical fair value of 54.0023. So we cannot make money trading futures against spot.Using our option-pricing model, we value calls and puts struck at the forward (i.e., at54.0023). We get 7.3887 for the put and 7.3887 for the call. We notice that we can buythe put in the market for 7.39 (the asking price), and we have a customer willing to pay7.42 for the call. One simple strategy is to sell the call to the customer for 7.42 and try tooffset that by buying it in the market for less. But suppose the market is wide at 7.38/7.42 (i.e., the asking price is 7.42): we can still profit by establishing a synthetic shortforward in the options and either going long futures or establishing a synthetic longforward in the cash markets (buying stock and financing the purchase at 2.6%). Let usconsider the first choice.We sell a call for 7.42 to the customer and buy a put for 7.39 in the market, both

struck at 54.0023. We go long one futures contract with the same expiry at 54.0023(assuming contract sizes or multipliers on the options and futures are the same). Ournet premium is 0.03 which is also our net profit. If the stock (and the futures) ends up at62 at expiry, we owe 7.9977 to the call owner, our put is worthless, and we collect 7.9977in the futures variation margin.If the stock ends up at 50, we owe nothing on the call, we collect 4.0023 on the put, and

we have lost 4.0023 in the futures variation margin. The second choice is a little morecomplicated. We sell the call for 7.42 and buy a put for 7.39 to collect a premium of 0.03.We borrow money at 2.6% per month to buy the underlying stocks for 50. In 3 monthswe sell the stock basket (inflow) and repay 54.0023 (outflow). This, as we have shownbefore, is equivalent to the cumulative variation margin on the long futures position. Theshort call–long put option position has the opposite payoff equivalent to the cumulativevariation margin on the short futures: the inflow of the strike price and the outflow ofthe value of the stock. The future cash flow is 0. Today’s cash flow is 0.03.Analogous strategies can be employed with commodity and currency options. We

compare the cost of replicating a forward position through options to the actualforwards/futures or their synthetic equivalents in cash positions.Options are sometimes used in index arbitrage in an attempt to better time program

execution and to recoup transaction costs. Suppose we want to go long futures andshort synthetic in large size. Selling a basket spot may take time and make force us tosell many stocks at the bid. Suppose, instead of buying futures, we leg into the trade bybuying a call and sending a sell order. If stock prices tick up, we may be able to getbetter execution on the program and still sell the put at the desired price. If bids in the

Option Arbitrage 277

cash market weaken, we may still be able to hit them and sell the put at a better price.Of course, the opposite could easily happen. Legging into a trade is always a small beton the execution costs.

Warrant arbitrage

Issuers in some markets often attach warrants to their bonds in order to make thebonds more attractive to investors. Warrants are call options that give the holders theright to buy the shares of the issuing company at a predetermined strike price. The onlydifference between warrants and standard calls is that the shares to be sold on exerciseof the options will be newly issued. This is possible in a warrant contract because thewriter of the option is also the issuer of the stock.1 Often these ‘‘bond sweeteners’’ comefree or close to free.

There are two distinct cases here. We illustrate the first with a fairly common situa-tion with Japanese industrial bonds in the early 1990s.

A Samurai Co. issues a 3-year bond to investors. The bond has a warrant attachedthat allows investors to buy Samurai stock for Y¼50 a share in 3 years. The yield on theSamurai bond is less than that on Samurai’s straight bonds to reflect the value ofattached warrants. We discover, by comparing the present value (PV) of the bond’scoupons and principal with the PV of similar bullet bonds, that the yield give-up (i.e.,the cost of acquiring the warrant) is equal to Y¼1. When we price a 3-year call using theBlack–Scholes or binomial model we compute the fair price of the warrant as Y¼10.0475.One way to profit would be to buy the bond with the warrant, detach the warrant, andtry to find buyers for the warrant at a premium close to the fair value. But other buyersmay have acquired the warrants at a low cost just like we have.

Here is the simplest strategy. We short the stock today. We collect Y¼50 from theshort sale and invest the proceeds at 2.6%. In 3 months we have:

50ð1þ 0:026Þ3 ¼ Y¼54:0023

If the stock ends up in-the-money, say at 60, we exercise the warrant by buying the stockat 50 and returning the stock to the lender on our short. We pocket Y¼4.0023. If the stockends up out-of-the-money, say at 45, we buy it back in the market for 45, return it tothe lender on the short, and pocket Y¼9.0023. The minimum we will make is Y¼4.0023.

In the 1990s, many outstanding warrant bonds issued by Japanese industrial com-panies offered this static arbitrage opportunity. The trade consisted of acquiring thewarrants which often had many years left to maturity, shorting the stocks, and waitingto collect. The only problem with pursuing the strategy was the availability of the stockfor shorting. Only those with access to large long-term holdings of the stock couldexecute the strategy. Insurance companies that held the stocks as investments lent themout, but often charged significant stock borrow fees. If in our example the borrow feefor the entire time to maturity were set to Y¼6, then the strategy was not risk-free. It lostY¼1.9877 if the warrant ended in-the-money.

Let us briefly describe the second case. We elaborate on this case when we discussconvertible bonds later in this chapter.

Instead of gambling to make Y¼4.0023 or more depending on where the stock ends up


1 This leads to the dilution of earnings from existing shareholders to new shareholders.

(minus any stock borrow fees), can we somehow lock in the difference between thecomputed fair value of Y¼10.0475 and the cost of the warrant of Y¼1? The answer is yes,but again only by shorting the stock.In the static case we shorted one full share per warrant. This time, we short only the

delta number of shares (close to half a share) and adjust the amount of shortingthroughout the life of the warrant (i.e., we dynamically replicate the payoff of thewarrant through a delta hedge). Our cash flows at expiry will be matched. The startof the delta hedge at the outset generates an inflow (short sale minus the money lending)of Y¼10.0475. We spend Y¼1 on the warrant and keep Y¼9.0475. The only thing thatcomplicates things is again the stock borrow fee which is non-zero, but lower than inthe static case (we start shorting only half the amount). The way the borrow fee is takeninto account is to treat it like additional dividends in the option-pricing model. Whenwe short stock we have to compensate the lender of the stock for the lost dividend. Thestock loan fee is paid on top of that just for the privilege of using the stock. We have touse an amended version of the Black-Scholes model to compute the warrant’s fair valueand all the deltas now and in the future.

10.2 RUNNING AN OPTION BOOK:

VOLATILITY ARBITRAGE

We explained in Chapter 9 that an option dealer hedges out the net price or rate risk ofhis option portfolio. He does this by faithfully following a delta-hedge recipe. Hecomputes the exposures of all options to the underlying asset or assets (i.e., pricesensitivities to a unit change in the underlying prices or rates), nets these acrossoptions, and takes the opposite positions in the underlying assets. Every day as theprices of stocks, currencies, and commodities or the rates on bonds change, the mark-to-market change on the options is exactly offset by the change in the value of thehedging assets. The dealer has to rebalance that portfolio daily to ensure that, asconditions change, the hedge assets will continue to offset the value of the optionportfolio.We also explained that the only risk the dealer is left with is that of the mismatch

between the implied volatility used as an input into the pricing and hedging model andthe actual volatility of the hedge assets. For example, if the dealer has a net shortposition in options (i.e., a net negative vega position) and the underlying asset pricesfluctuate more than he had assumed, then his hedge will have a tracking error. It isimportant to realize that this error is small relative to the completely unhedged position.At the same time, if the option portfolio is very large, it can amount to substantiallosses over time.The dealer’s job can be viewed as having two parts: one is fairly mechanical and

involves computing the exposures every day and rebalancing the hedge portfolioaccording to a recipe; the other part is dynamic and involves forecasting the path ofthe volatility of the assets and taking pre-emptive action.

Hedging with options on the same underlying

The only perfect hedge for an option is another option. If we have an option positionthat is already hedged in the underlying asset, the only way we can offset the volatility


exposure of that position is by buying or selling another option with a similar vega. Wewill illustrate this principle using our one-step binomial applied to a three-optionexample.

Suppose as an option dealer, we have lots of bought and sold options in ourportfolio, all written on the same underlying stock. The net exposure of that portfoliocan be summarized as equivalent to a short 1-year call option struck at 60 and two short1-year put options struck at 50. We assume the current stock price is 50 and the interestrate is 10%. The up and down states, which reflect the current implied volatility, are 70and 20 as before. We price the options using those inputs and also using the 80–15inputs, reflecting a higher implied volatility. The results are as follows.

Figure 10.1 Pricing options with two volatility inputs to get the vega.

We have one short call in the portfolio worth 6.3636 and two short puts each worth8.1818. The vega of the options is equal to the difference between the price under theincreased implied volatility assumption minus that under the current one. (In reality,stock options traders plug the inputs into their Black–Scholes models and revalue theoptions using an implied volatility increased by 1% to determine their vega exposure.)For the call, we have:

11:1888� 6:3636 ¼ $4:8252

and for the put:

12:2378� 8:1818 ¼ $4:0559

We also show the deltas for each option: 0.20 for the call and -0.60 for the put. Supposeour portfolio is delta-hedged so that our combined option-and-hedge holdings are:


Table 10.2 Delta and vega of the position

Position Security Unit PV PV Unit delta Delta Unit vega Vega

�1 60 Call 6.3636 �6.36364 0.20 �0.2 4.8252 �4.82517�2 50 Put 8.1818 �16.3636 �0.60 1.2 4.0559 �8.11189�1.00 Stock 50 �50 1 �1 0 0

——————— —— —————Net �72.7273 0 �12.9371

Since our combined delta from the three options was 1.0, we short one share to hedgethe option exposure (we should have bought 0.20 to hedge the call and shorted 0.60 tohedge each put). We have a net vega exposure of �12.94 (i.e., if the volatility increasesas expected, the short options will provide us with a mark-to-market loss of 12.94).In order to hedge the volatility risk, we look for positive vega exposures in the

market. Suppose we can buy 55 puts in the market for 9.30. We use our pricingmodel to verify.

Figure 10.2 Computing the vega of the 55 put.

We apply the model twice with different volatility inputs to compute the vega of the 55put to be:

13:986� 9:5455 ¼ 4:4406

We also notice that the puts are cheap relative to their fair value. We decide to hedge bybuying some 55 puts. How many? We need to offset 12.94 in volatility exposure; so,given the vega of each put of 4.4406, we need to buy 2.9134 puts. But if we buy 2.9134puts, then we need to delta-hedge them by buying:

2:9134� 0:70 ¼ 2:0394

shares. Our combined portfolio now looks like this:


Table 10.3 Delta and vega of the hedged position

Position Security Unit PV PV Unit delta Delta Unit vega Vega

�1 60 Call 6.3636 �6.363 64 0.20 �0.2000 4.8252 �4.825 17�2 50 Put 8.1818 �16.363 6 �0.60 1.2000 4.0559 �8.111 892.9134 55 Put 9.5455 27.809 73 �0.70 �2.0394 4.4406 12.937 131.04 Stock 50 51.968 5 1 1.0394 0 0

————— ———— —————

Net 57.050 96 0.0000 0.000 1

If the volatility changes, the change in the value of the extra puts will offset the changein the value of the original options in our portfolio. Notice that this will work for avariety of volatility changes (i.e., even if the volatility decreases instead of increasing).

Volatility skew

Our primary risk (to the change in the price of the stock) and our secondary risk (to thechange in the volatility of the stock price) are now eliminated. Is our portfolio risk-free?

No, our portfolio can still make or lose money if the implied volatilities on thedifferent options change differently. We are left with a tertiary risk of relative volatilitychanges. Theoretically, this should not happen as the underlying asset for all options isthe same and so it can have only one true volatility. But our model is not perfect. Inparticular, it assumes that volatilities do not change! It also assumes that volatilities arethe same for any asset price. The volatility skew intends to correct for that by assigningslightly different volatilities for different strikes of the options (something we did notdo).

Let us illustrate what happens when a dealer fails to include the skew in hedgecalculations. We focus solely on the vega. We use the data from Chapter 9 for Decem-ber, 2003 S&P 500 index futures options as of October 15, 2003. With the index at 1045,we show the implied volatilities for strikes between 1030 and 1075, and a fitted regres-sion line Vol ¼ 57:638� 0:0388 � Strike.

Figure 10.3 Volatility skew of S&P 500 (fitted vs. actual).


Table

10.4

Hedgeperform

ance

withoutandwithskew

Strike

Before

movewithoutskew

After

movewithoutskew

Before

movewithskew

After

movewithskew

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

——

———

——

———

——

———

——

——

——

——

———

——

———

——

——

Implied

Vega

Position

Start

vega

Implied

Vega

Position

Endvega

Implied

Vega

Position

Start

vega

Implied

Vega

Position

Endvega

volatility

per

unit

volatility

per

unit

volatility

per

unit

volatility

per

unit

1030

17.00

1.69

�30

�50.7

17.00

1.71

�30

�51.3

17.72

1.71

�30

�51.3

17.51

1.73

�30

�51

1035

17.00

1.71

0.0

17.00

1.73

0.0

17.51

1.73

0.0

17.28

1.74

0.0

1040

17.00

1.73

0.0

17.00

1.75

0.0

17.28

1.74

0.0

17.04

1.75

0.0

1045

17.00

1.75

100

175.0

17.00

1.73

100

173.0

17.04

1.75

100

175.0

16.90

1.75

100

175

1050

17.00

1.73

0.0

17.00

1.71

0.0

16.90

1.75

0.0

16.65

1.74

0.0

1055

17.00

1.71

0.0

17.00

1.69

0.0

16.65

1.74

0.0

16.50

1.73

0.0

1060

17.00

1.69

0.0

17.00

1.66

0.0

16.50

1.73

0.0

16.30

1.70

0.0

1065

17.00

1.66

�42

�69.7

17.00

1.63

�42

�68.5

16.30

1.70

�40

�68.0

16.16

1.67

�40

�66

1070

17.00

1.63

�33

�53.8

17.00

1.59

�33

�52.5

16.16

1.67

�33

�55.1

15.97

1.62

�33

�53

1075

17.00

1.59

15.97

1.62

Total

0.8

0.8

0.6

2

In Table 10.4, we illustrate what happens to a dealer who does not take the volatilityskew into account. We assume that we start with the index at 1045 and compute the at-the-money volatility to be 17%. We own 100 options struck at-the-money at 1045. Wehedge the volatility risk by selling options struck at 1030, 1065, and 1070. If we do notinclude the skew, then we will assume the options with out-of-the-money and in-the-money strikes have the same volatility. We compute their vegas using that volatility;this is how we come up with the number of options to be sold to hedge. If we take theskew into account, then we come up with a different set of vegas (those in the originaldata) and different numbers of options to be sold. We also show what happens as themarket moves. Suppose the index goes down to 1040. We assume that the impliedvolatilities and the vegas do not change for the same percentage of in-the-moneyness.In our case, that translates, for example, into the 1050 option having now the sameimplied volatility as the 1055 option had before the move. We see that a dealer unawareof the skew continues to think that he is hedged and does not rebalance, while a dealeraware of the skew sees that he is no longer vega-hedged and will most likely sell moreoptions.

What we did not show in Table 10.4 is that not only will the volatility risk be mis-hedged, but also the deltas will be computed using the wrong � input. The dynamicdelta hedge will result in daily unexplained profit or loss (i.e., the change in the deltaposition in the stock will not match the change in the value of the options).

Most professional skew models do not rely on a simple regression model like ours tomodify the volatilities used in computing prices and hedge ratios in a Black–Scholesmodel. Rather, they internalize the volatilities as stock level-dependent within oneoption-pricing model, used with the same inputs, to price all options with differentstrikes. This is equivalent to having different volatilities on different branches of ourbinomial trees in the up and down direction. This ensures the consistency of both delta(stock position) and vega (option position) hedges.

Options with different maturities

The comments about options with different strikes apply to options with differentmaturities. The implied volatilities of options on the same underlying, but with differentmaturities, are not the same and will change differently over the life of our portfolio.Dealers find it very hard to hedge long-expiry options with short-expiry options eventhough that is what they are most often asked to do by customers who can buy short-term options on the exchanges and come to dealers for longer term or more complicatedbets. The implied volatilities on long and short options can be quite different. Longeroptions’ deltas are also less volatile, so the hedge with short options requires moretrading of the underlying. Dealers find themselves trading off the transaction costs ofthis strategy against potential benefits of the hedge.

10.3 PORTFOLIOS OF OPTIONS ON

DIFFERENT UNDERLYINGS

If a portfolio contains options written on different underlying assets—some on bonds,some on stocks, and some on currencies—then each option has to be delta-hedged


separately. But the notion of ‘‘different underlying’’ need not be so clear-cut. Weconsider a few cases where underlying assets are different, but at the same timesimilar. In these cases, the underlying assets are not completely independent of eachother; they exhibit co-movement. Their volatilities can be postulated to be statisticallyor mathematically related.

Index volatility vs. individual stocks

Instead of a portfolio of options all written on the same stock, suppose we have aportfolio of options on a variety of stocks. Each option is delta-hedged with a positionin its own underlying stock. Can we do anything about volatility exposure?The only method that will work well is what we already described in the previous

section. We have to find vega offsets for options on different underlyings one at a time.Buying options on Intel in order to hedge vega exposure on Microsoft can be adangerous game. Ideally, we need to find options on Microsoft. But hedging all ex-posures separately can be prohibitively expensive. We may soon find out that we haveto pay ask prices well above fair values on a multitude of individual options, incurringan instant mark-to-market loss. So, to hedge a portfolio of options we may be temptedto look for an option on the entire portfolio or an index option.Suppose we have bought and sold a variety of options on 35 individual stocks in the

S&P 100 index. We have a net short-vega exposure. So our thinking is that when theimplied volatilities of the individual options rise and we stand to lose money, we canprotect ourselves by buying an option with the same vega on the entire index. To alimited extent, this may work; but, in general, it will amount to a substantial bet on thecorrelation structure of the index. If the stocks in the index are highly positivelycorrelated (say, they are from the same country or industry), then most likely assome stocks increase the others increase and as some stocks decline the othersdecline too. So the average volatility of the stocks is related to the volatility of theindex. But the change in average volatility may not be so clearly reflected in the changeof index volatility if the stocks in the index are not highly positively correlated; this iswhat we are trying to hedge. The hedge breaks down when the average volatility of thestock changes by 2%, while index volatility changes by 1%. We can try to estimate thatdifferential and adjust for it, but the game is risky.The strategy of hedging individual options with basket (index) options is highly

speculative. The situation is similar to that with volatility skew exposure acrossstrikes and maturities. While we know how to translate a volatility input into anoption value and a delta (Black–Scholes model), we do not know the exact relationshipbetween the volatilities within the skew (regression?). All the options within the skeware on the same underlying asset and therefore, over large movements of the under-lying’s price, they have to be highly correlated. Because of this high correlation, as weargued, the average volatility in the skew portfolio will move closely with the volatilityof any option close to the money. This need not be the case with options on differentstocks relative to average volatility and relative to the option on a basket: the reason thehedge is more speculative in this case.


Interest rate caps and floors

Portfolios of caps, floors, and certain exotic options can all be viewed as all options onshort-term interest rates with different maturities. The short-term rates may be 1-, 3-,and 6-month LIBOR (London interbank offered rate) or other short rates (e.g., in theU.S., Fed Funds rate or T-Bill rate) related through a ‘‘spread’’. Each cap and eachfloor is a portfolio of caplets and floorlets seemingly on the same underlying rate (say,3-month LIBOR). But is the 3-month deposit today the same asset as a 3-month deposit4 years from today? The implied volatilities in the caplets of increasing maturities can bedifferent and change differently over time. But they are related.

The procedure of cap volatility curve calibration, which relies on bootstrapping, isportrayed in Figure 10.4. Suppose we view the spot 3-month LIBOR as a random assetsimilar to a stock, subject to a volatility parameter moving through time, or a binomialtree. We observe caplet prices with 3-month, 6-month, 9-month, etc. maturities. Fromthe 3-month caplet price we can back out an implied volatility. We can then considerthe 6-month caplet. We let LIBOR fluctuate for the first 3 months with a volatility ascomputed in step 1, and then we imply what the volatility for the following 3 monthswould have to be so that the model price of the caplet would match the actual price. Wethen consider the 9-month caplet using the first two volatilities, for 0–3 months and for3–6 months, and implying the 6–9 month one. In a binomial tree model we are effec-tively computing different volatilities for different time steps (left to right).

Figure 10.4 Cap volatility calibration. Term vs. local volatility.

Computed volatilities are often referred to as local volatilities or forward–forwardvolatilities. The single volatilities that would be required in a model to obtain thetrue prices are referred to as term volatilities. These represent the total width of thedispersion for a given point in time. Local volatilities represent the dispersion relative tothe previous step. In Figure 10.4, we deliberately included one odd outcome, where theterm volatility shrinks relative to a prior period. This is not unusual. We often find that


long-term volatilities are lower than short-term ones. What that necessarily implies isthat the local volatility for that period is negative or, alternatively, that the model hasmean-reversion (i.e., outcomes are constrained from above and below).The simplest procedure to deal with the volatility exposure for a cap portfolio is to

disassemble it into caplets and compute the term vega exposures of all the options to aset of benchmark caplets with different maturities. These can then be combined to forma set of benchmark caps, those with the most liquidity in the inter-dealer market. Wecan then see where the bulk of the exposure is and design the start date and end date ofcaps or floors that would hedge the computed exposure. An analysis of an averagestrike for each caplet also helps select the strike of the hedging cap to minimize skewrisk.

Caps and swaptions

We might think that the logic of cap and floor-only portfolios can apply to hedgingentire, fixed income derivative portfolios. After all, swaptions are options on longer-term rates, and long-term rates can be replicated from forwards on short rates. So,swaption and cap volatilities should be closely related. This could not be further fromthe truth. The yield curve changes continuously. Long and short rates do not movetogether and are not equally volatile. A long par rate is a function of short-term ratesover time. However, the volatility of the long rate is a function not only of the impliedvolatilities of forward short rates, but also the correlations between them. The bestanalogy here is with stock index volatility relative to individual stock volatilities.Hedging caps with swaptions and vice versa suffers from the same problems (i.e., itis a bet on the correlation structure of the rates). As much as it is clear with indices andstocks, the issues in swaptions and caps are often befuddled because the pricing modelsin interest rates are very complicated and at the same time not complicated enough.They try to capture all the synthetic static arbitrage relationships between rates(coupons vs. zero-coupons, short vs. long) while simplifying the volatility structure toone or at most two factors (i.e., they are arbitrage-free, but perhaps only one-factor).The simplification often leads to the omission of the rate correlation structure. Even aseemingly complicated model with both local and term volatilities can be one-factor ifthe only thing driving rate movement is the short-term rate. Any long-term rate for anynode on a model tree is simply an arbitrage-free deterministic outcome of a present-value calculation conditioned on being at that node. Model purists often object to anyviolations of no-arbitrage rules. Yet allowing some arbitrages might allow easier multi-factor modeling which can then capture the calculation of the exposure to correlations.That knowledge allows us to design more complicated spread hedges consisting of bothcaps and swaptions with the same combined volatility and correlation exposure as theunderlying portfolio.At the present time, many dealers combine the delta hedges (in the underlying

deposits, futures, and bonds) for cap and swaption portfolios, but run the volatilityrisk on cap and swaption portfolios separately, hedging the first (caps) exclusively withoptions on short-term rates (caps) and the latter (swaptions) with options on long-termrates (swaptions or CMSs/CMTs). They monitor the total vega of each portfolio anddo so relative to each other. Frequently, they leave the vega of each part, viewed


separately, unhedged, betting on the convergence or divergence of cap and swaptionvolatilities.

Explicit correlation bets

Option traders in many markets buy and sell bets whose primary risk (after delta-hedging the underlying) is the correlation risk. These bets are known as spreadoptions or best-of options.

In equity markets, option sellers offer spread options on the difference between theprices of two stocks or indices; that is, the payoff is defined as:

Max½Stock1 � Stock2 � Strike; 0�If the two underlyings are stock indices for two different national markets, then theseoptions may be offered with quanto features. A different structure, exposing the dealerto the correlation between stocks or indices, has a ‘‘best-of ’’ payoff:

Max½Index1 � Strike; Index2 � Strike; Index3 � Strike; 0�The owner of such a ‘‘call’’ is entitled to the return on the best of three indices. Apartfrom the current levels of the indices, the primary variable driving the price of theoption is not the volatility of the indices, but their correlation structure. Similarspread structures can be written on two or more currency rates.

In commodity markets, option traders typically offer spread structures on two seem-ingly related commodity prices. These can be, for example, oil price and jet fuel price.Customers (airlines) demand such options to hedge the price volatility of the inputs intotheir business cost structure ( jet fuel). As futures contracts on the commodity of interestmay not exist ( jet fuel), they hedge with proxy instruments (oil) by incurring basis risk.The spread structures allow them to close the hedging gap. But structures like these mayleave the dealer with a speculative rather than a hedged position, if one or both assets inthe basis do not have futures contracts on them. The dealer’s job is then to protecthimself by running a matched or diversified book.

In interest rates, the most common version of the spread option is that written on twodifferent points on the yield curve. The payoff may then read:

Max½10-year rate� 5-year rate� Strike; 0�or

Max½5-year rate� 10-year rate� Strike; 0�The first option can be thought of as a bet on the steepening of the yield curve, or a‘‘steepener’’, as it pays when the par 10-year coupon rate rises by more that the par 5-year coupon rate (or declines by less). The second option is a ‘‘flattener’’. It pays whenthe yield curve flattens (i.e., the 10-year rate rises by less than the 5-year rate).

In all these cases, the correlation exposure is unhedgeable unless there exist spot orforward instruments on the spread itself. The delta hedge can be easily constructed bybuying one asset and shorting the other, but more often than not it provides insufficientprotection against unpredictable relative moves.


10.4 OPTIONS SPANNING ASSET CLASSES

There exist cash securities and related derivatives that cannot be neatly classified asbelonging to equity, fixed income, or currency markets. They present holders withunique risks of the interaction of prices of and secondary exposures to assets in differentmarkets.There are assets that appear to belong to this group, but when dissected are actually

structured securities built from separate components. A bond with an equity warrantattached to it is really two separate assets that have nothing to do with each other, eventhough the yield on the bond part may be lower to cover the cost of the warrant. Thetwo components can, however, be priced and hedged separately. A structured bondwhose coupons are tied to the performance of a stock index, rather than an interest rate,can be easily decomposed into a string of stock index forwards or options and a set ofzero-coupon bonds. Each can be priced separately to give the total price of the package,but each is then hedged separately.However, there are certain assets that cannot be dissected into separate components.

Convertible bonds

The most common example of a security spanning asset classes is a convertible bond.When the embedded option is exercised, the bond is tendered. The holder gives up theright to further coupon interest in exchange for shares. His decision to exercise, whilemost often is prompted by the rise in the stock price which makes the exercise value(stock) greater than the value of the unexercised bond, can be prompted by the declinein value of the bond coupon stream due to interest rise. The embedded equity optioncannot be priced as a straight call. In addition, a convertible bond may be callable,which makes the evaluation of the bond even more complex. The call provision fre-quently allows the issuer to force reluctant bond holders to exercise the equity option. Ifthe intrinsic value of the equity option is above par, then the holder would, uponexercise, receive shares valued at more than par (a call by the issuer at par willprompt the holder to exercise in order to avoid the loss of value). The issuer’s decisionto call bonds may be driven not by equity considerations, but by a decline in interestrates and the desire to refinance debt at lower rates.These considerations have to be taken into account in the pricing/hedging model. The

volatilities and the correlation structure of the stock price and the interest rate(s) haveto be modeled explicitly. The daily rebalancing of the hedges also has to take intoaccount the movement in all related asset classes. A change in interest rates mayentail a change in the position in the underlying stock and vice versa.We describe the valuation problem for convertibles with the following diagram of a

two-dimensional binomial tree. The first dimension is the interest rate, and we have upand down nodes based on its volatility. The other dimension is the stock price, and wehave up and down nodes based on its volatility. We also have another condition thatbinds all the nodes: the correlation of the interest rate and the stock.


Figure 10.5 Two-dimensional pricing grid for convertibles.

We omit the math of setting up the tree. The main logic of the valuation is that, as wesweep backward in the tree, we apply the exercise condition in order to compute thevalue of the convertible bond on that node. The condition compares the exercise valueof the bond (number of shares into which it is exercisable times the price) with thediscounted expected value of the forward nodes. This is where the interaction of thestock price and the interest rate takes place. The discounted expectation includes allfuture coupon payments discounted to today (and all future possibilities of exercise).The delta with respect to the stock and the delta with respect to the interest rate thatcome up from the model are the product of the interaction of the two random variables.The model price of the convertible is equal to the amount needed to start the dynamichedge (to be rebalanced until the maturity of the convertible), where the hedge consistsof positions in the stock and in a purely interest rate-sensitive instrument (e.g., a bulletbond).

What the convertible model highlights is that, in practice, portfolios of bonds con-vertible into different stocks are hedged with baskets of stock and a variety of interestrate-sensitive instruments. Each bond has a delta with respect to its stock and a deltawith respect to interest rates. Stock deltas call for hedges in the underlying stocks, whileinterest rate deltas call for hedges in bonds. The latter actually involve hedges in risk-free as well spread-sensitive bonds (i.e., interest rate deltas can be broken down intoexposures to pure interest rates as well as corporate spreads).

More importantly, all the deltas are related to each other. A change in interest ratesmay require the hedger to change his stock position, and a change in the stock pricesmay require the hedger to change his bond position. The greatest danger of the hedgingscheme is that the real movements in stocks and rates do not reflect the assumedstatistical correlation in the model and the hedge slips (i.e., results in extra profit orloss).

Quantos and dual-currency bonds with fixed conversion rates

Any multi-currency bond that gives the holder a right to receive coupon interest in thecurrency of his choosing, with the foreign exchange (FX) conversion rate which is


predetermined, exposes the issuer to intertwined interest rate and currency risks. If,during the life of a bond, the value of a currency declines, the bond holder may chooseto have interest paid in that currency to take advantage of the rate originally contractedfor and not equal to the current market rate. The risk here is the same as in quantooptions where a payoff is translated into a non-native currency. A stock index quantocall gives the owner the payoff of a stock index not in the currency of the stocks in theindex, but in another currency at a fixed rate. The writer-hedger of the option is exposedto the correlation of the FX rate and the index level. As the index increases his FXhedge may increase.

Dual-currency callable bonds

A single-currency callable bond is equivalent to a straight bullet bond and a callswaption. The exercise of the call swaption leads to cash flows that exactly offsetthose on the bond, effectively canceling the bond. The issuer normally issues a callablebond and sells the swaption to a dealer, the sole purpose of the structure being a lowercost of capital.A single-currency callable bond in one currency swapped into another currency or a

dual-currency bond with conversion rights which is callable carries the correlation riskbetween the FX rate and long interest rates in the two currencies.

10.5 OPTION-ADJUSTED SPREAD (OAS)

Many bonds have options embedded in them. They are sold as packages of bullet bondsand options. Sometimes these options are physically separable, as in warrant bonds.Sometimes they are not physically separable, but the components can be treated separ-ately for valuation and hedge calculation purposes. A callable bond gives the issuer theright to call (repay) the bonds prior to maturity at par (or another call price). The buyerof the bond cannot be sure when he will be repaid the principal (at the latest atmaturity) and how many coupon payments he will get. Even though this option isembedded, it can be treated separately. A callable bond can be viewed as a bulletbond plus a call swaption with the strike equal to the coupon of the bond sold bythe bond holder to the issuer. When the bond is called, the remainder of the cash flows(which will be 0) can be constructed from the coupons on the bullet bond and theexactly offsetting payoff on the swaption. A puttable bond gives the holder the rightto put (redeem) the bonds back to the issuer at par (or another put price). A puttablebond can be viewed as a bullet bond plus a put swaption sold by the issuer to the bondholder. Callable bonds and puttable bonds can be priced and hedged by looking at theircomponents separately and netting the prices and deltas.What if the bond is both callable and puttable? Then the options cannot be treated

separately, because the decision to call depends on whether the bond has been put andon the future possibilities of being put. Vice versa, the decision to put depends on priorand future decisions to call. The two are intertwined. This is identical to the convertiblebond situation where the decision to exercise the conversion option depends on futureshare prices and interest rates together. In Chapter 8, we discussed structured financewith the use of derivatives like swaps and caps. A callable inverse floater is an example


of a bond with embedded options that are not separable. The inverse floater’s couponcontains a cap protection as the coupon rate cannot fall below 0. On top of that, theissuer can call the entire package and his decision will depend on whether the caps areto be exercised or not.

In these and other examples, the embedded options are dealt with using the rightpricing model (i.e., one that does not separate the components, but values and com-putes delta hedges for the whole package). To provide information to investors, dealersoften compute and quote measures of optionality in bonds. The option-adjusted spread(OAS) is defined as the difference between the yield on the structured bond minus theyield the bond would have if all the embedded options (whether separable or not) wereeliminated.2 For example, for a convertible bond, the OAS is equal to the yield on theconvertible minus the yield on an equivalent straight bullet bond with the same matur-ity. OASs are quoted for most bonds we have already mentioned.

They are also very popular in mortgage-backed securities (MBSs). Let us consider thesimplest example of an MBS: a fixed-coupon pass-through. A pass-through is a bondwhose coupon stream matches the payment stream of the underlying pool of mort-gages. Each month a pass-through holder receives coupon and principal repayments.The biggest risk of fixed-coupon pass-throughs is the embedded option sold by thelender (bond holder) to the borrower (mortgagee). Any homeowner has the right toprepay the mortgage by paying it in full (e.g., when selling the property) or sendingextra checks to reduce the principal. Each homeowner may have his own reasons forprepaying, but, in general, he will tend to prepay when interest rates are low (in order torefinance). The pass-through owner will be subject to the totality of the prepaymentdecisions coming from the entire pool. The embedded option is equivalent to a callswaption held by the homeowners who can call their loans prior to maturity. Theoption is not exercised optimally as soon as interest rates go down, as it depends onthe sum of individual decisions.

Just as callable bonds have OASs quoted on them, so do MBSs. This allows investorsto compare the level of uncertainty about the timing of the cash flows, all subsumedinto one summary statistic. We need to remember that this statistic, however, hides a lotof detail about the exact reasons for exercising the embedded options. This is particu-larly important for MBSs where the prepayment speeds are just estimates. In contrast,true callables and convertibles are exercised rationally whenever the exercised valueexceeds the unexercised value.

Just like large bond portfolios, which are often characterized by average durationand convexity numbers, portfolios of bonds with embedded options can be described byaverage OASs on like securities. These however are not additive and cannot be used tocome up with heuristic hedges. Individually, they provide crude clues to volatilityexposure.

10.6 INSURANCE?

The last section of this chapter is devoted to the discussion of financial contracts thatlook like other financial options; but, because they are truly unhedgeable, they are more


2 Sometimes the definition is in terms of the spread over the spot model (discount) rate (rather than the quoted yield) whichcan value the bullet and the structured bond to the observed prices.

akin to insurance products than options. Insurance companies do not hedge their risks;they cannot take positions in underlying assets as most of the bets they sell are onevents. Instead, they manage the risks through diversification and reserve management.They avoid higher probability events or accident-prone individuals; they spread risksacross ages, sexes, types of liability, geography, etc.; and they build reserves in goodyears to use in years with high payouts. They do not manufacture payoffs the wayoption dealers do. Option dealers look to find underlying positions that dynamicallytraded can eliminate most risks. Sometimes that is not possible and they venture intoinsurance-like bets. This is an ever-growing segment of the market as financial engineerspush the envelope of financial innovation.We start with the simple example of an option-insurance borderline case of long-term

options.

Long-dated commodity options

Suppose we sell a standard call option on the price of petroleum. Let us choose Brentcrude oil whose current spot price is $28 a barrel and set the strike price equal to $35.The hedging of commodity options with spot trading is impractical and expensive.

Buying oil outright would lead to high storage costs and could only be done for alimited quantity. If we had sold a put, we would have to short oil spot which would beimpossible. The standard way to deal with these problems is to buy or sell futures withthe same maturity as the option expiry. The futures will converge to spot by the expirydate, and so the futures is the perfect hedge tool. It can be bought and shorted, andrebalanced quickly and with minimal transaction costs.We set the expiry of the call option we sell to be 10 years. However, futures contracts

go out at most 2 years. We are subject to unexpected supply shock risks that drive awedge between the fair value of futures without the convenience yield and the actualprice. Oil futures are subject to normal backwardation as we explained before. That is,each futures contract price deviates from its fair value based on today’s spot, thefinancing cost and storage costs, and the convenience yield markup, reflecting noassurance of delivery of a physical commodity. So if we dynamically hedge with thetraded futures, our cost of manufacturing the payoff will be a function of the conve-nience yield which fluctuates in an unpredictable fashion. Any supply shocks (war,supply gaps) will exacerbate the fluctuations of the futures price from fair value.None of this would matter if we had a 10-year futures contract, except for the potential‘‘jumpiness’’ of its price. But because there is no 10-year contract, we have to hedgewith 1- or 2-year contracts which will be subject to different convenience yields. We willroll over the hedges on their expiry, or earlier, into new 1- or 2-year contracts (this isalso called stacking). In addition, the realized volatility of the spot price over 10 yearsmay be much lower than the volatility of the hedge. The estimated manufacture costbased on a hedge rollover assumption (with implied volatility) will be just as inaccurateas the cost based on the spot price (with historical volatility or some other estimate). Ineffect, if we engage in the hedge, we are taking an outright bet on the convenience yieldsand historical volatility; if we do not engage in the hedge, we are taking a bet on the oilprice itself. Often both choices are equally risky.


Options on energy prices

Another example of unhedgeable options—options on electricity prices—comes fromthe world of commodities as well. This time the spoiler is the fact that electricity cannotbe stored easily and has to be delivered to a specific point on the grid. While thecapacity to generate the spot product may exist, the ability to deliver it may not.

Suppose we sell an option on the spot price of electricity delivered to Los Angeles ona specific date at a specific peak time. We hedge with short-term futures contracts on theprice of electricity at the California–Arizona border. In deregulated or partially deregu-lated markets, electricity prices for two different points on the grid are not goodsubstitutes of each other. While they may appear to reflect the spot price of the samecommodity, the supply conditions (weather-related demand, grid overload, local gen-eration failures, etc.) may cause the prices to diverge dramatically. In short, a true hedgeinstrument does not exist. Although we may euphemistically refer to hedging with asubstitute as ‘‘basis risk’’, we may not have a hedge at all. Not only is there noassurance of the convergence of the two prices, the two may fluctuate differently.

Options on economic variables

Dealers are sometimes pushed by their customers to buy or sell options on economicvariables like the inflation rate, GDP level, or unemployment rate. Financial institu-tions that have sold client contracts with indexed payoffs (pension, health insurance)have indirect, but natural aggregate demand for such structures. Although the payoffson such products, for example:

Max½U.S. consumer price index� 2%; 0�may look like those of standard call and put options, these are not options at all; theyare essentially unhedgeable bets. Unless the dealer can find the other side of the trade totransfer the risk to (i.e., to reduce his role to a matched-book dealer—read: a broker),he is an insurance provider. Any hedge relying on some ‘‘basis risk’’ will be faulty. Amacroeconomic model may provide a statistical relationship between the underlyingrate (inflation) and some instruments (stock index, real estate index, long bond rate)which can be traded. The model will be unstable and, as a result, of limited use.

A final word

How are such products different from the previously covered spread options? The maindifference is that here there is no underlying price or rate, while the spread option hastwo rates or prices of well-defined underlying assets. The basis risk of insurance-likeproducts may sometimes be no larger than the correlation risk of the seemingly hedge-able options. In most cases, common sense is all that is needed to separate dynamicarbitrage from outright speculation.


__________________________________________________________________________________________________________________________________________ Appendix __________________________________________________________________________________________________________________________________________

___________________________________________________________________________________________________________________________________ Credit Risk __________________________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________________________________________________________ 11 ______________________________________________________________________________________________________________________________________________________________________

Default Risk (Financial Math IV) and

_________________________________________________________________________________________________________ Credit Derivatives _________________________________________________________________________________________________________

Up to now, we have assumed that all interest-bearing securities were default-free. Theprices (present values) of deposits, bonds, and swaps varied as interest rate changed, butthe cash flows promised from these contracts, even those unknown at the outset(LIBOR-indexed interest), were sure to be received at the scheduled dates. Alterna-tively, for securities with non-guaranteed cash flows (stock dividends), we have assumedthat the interest rate charged reflected the riskiness of the investment. For some transac-tions, we assumed that the interest rate reflected the quality of the collateral (repos) orthe credit standing of the guarantor (lending against the box). In most of the discussionso far, we ignored any issues related to compensation for the risk of default (i.e., willfulnon-performance on the contract).It is, however, very important to understand how the possibility of default is factored

into the yields on fixed income securities. In general, the yield on any interest-bearingsecurity can be decomposed into two parts. The first reflects the general cost of fundsfor a given maturity (short maturity vs. long maturity) and type (zero, coupon, oramortizing interest). This is due to supply and demand forces in money and capitalmarkets for given maturity ranges. The second reflects the possibility that the issuer ofthe security will fail to pay one or all of the promised cash flows. This can be due to theshort-term liquidity problems of the issuer or due to bankruptcy. Credit-rating agenciesspend countless resources researching and analyzing all relevant information to assessthat probability of failure to perform. The prices of fixed income securities change as aresult of changes in either of the two components. In most of this chapter, we willassume that the first component does not change (i.e., default-free interest rates stay thesame). The only part that changes is that due to the changing probability of default.The default risk of the issuer is reflected in a credit spread. The credit spread is defined

as the difference between the yield on a default-risky security and a default-free securityof the same maturity and type. Any security issued by the government of the currencyof denomination can be used as a default-free benchmark. One simply matches thematurity of the reference rate to the given security. For example, if the yield to maturity(YTM) on a 5-year U.S. dollar-denominated coupon bond issued by ABC Corp. is6.35% and the YTM on a 5-year U.S. Treasury is 5.84%, then the credit spread for5-year ABC bonds is said to be 51 basis points (bp).This chapter consists of two main parts: sections 11.1–11.4 cover the mathematics

(Financial Math IV) related to default risk; and Sections 11.5–11.7 provide an overviewof credit markets arbitrage, particularly with the use of credit derivatives. We start our

discussion of credit spread by developing a simple model of how a probability of defaulttranslates into spread.

11.1 A CONSTANT DEFAULT PROBABILITY MODEL

Let us examine a 3-year annual bond issued by ABC Corp. The bond carries a 5%coupon. We also observe the following rates on default risk-free securities.

Table 11.1 Rates of default-free bonds

Risk-free—————————————————————

Year Forward yield Zero yield Par yield

1 2.50 2.500 000 2.500 0002 2.75 2.624 924 2.623 3053 3.00 2.749 797 2.745 279

Suppose we assume that the probability that ABC will go bankrupt over the next year is0.11%. We further assume that if ABC goes bankrupt then not only will it not pay the5% coupon due in 1 year, but it will also fail to pay any coupons after that. If, however,ABC survives the first year, then the probability that it will go bankrupt in the secondyear will again be 0.11%. If ABC fails in year 2, then it will fail to pay the coupon inyear 2 and the coupon and principal in year 3. If it survives year 2, then it will again facethe probability of default of 0.11% for year 3. For each year the survival probability isconstant at 99.89%.

What follows is that the cumulative survival probability up to any given time (oryear) n is equal to 0.9989n. What also follows is that the expected present value (PV) ofany coupon payment is equal to the cumulative survival probability times the promisedcash flow (CF) times a discount factor (DF), where the discount factor is computedusing default-free rates. This is because if ABC does not survive to pay a given cash flowthen its PV is equal to 1 minus the cumulative probability of survival times 0 as the cashflow times some discount factor, or 0. The analysis is summarized in Table 11.2.

Table 11.2 PV of ABC’s bond in a default probability model

Year CF Probability of Cumulative Risk-free PVdefault per year survival probability ———————————(%) Zero rate (%) DF

1 5 0.11 0.998 90 2.5000 0.975 610 4.87272 5 0.11 0.997 80 2.6249 0.949 499 4.73713 105 0.11 0.996 70 2.7498 0.921 843 96.4745

Sum PV ¼ 106.0842

Yield ¼ 2.8550%

Risk-free yield ¼ 2.7417%


The promised cash flows in years 1, 2, and 3 are 5, 5, and 105, respectively. We use therisk-free zero rates to discount the cash flows if they are received. The PV of each cashflow is the expected value computed as the weighted average of two possibilities: thefirst, weighted by the cumulative survival probability, is the PV of receiving the cashflow; and, the second, weighted by the remaining probability mass, is 0 (i.e., the PV ofnot receiving a cash flow). The price is 106.0842. Given that price, we can back out theYTM on the bond to be 2.8550%. Given the risk-free zero-coupon rates, we cancompute the default-free yield on 3-year coupon bonds as 2.7417%. The credit onABC’s bond is 0.1133%.Suppose ABC’s rating changes to ‘‘junk’’ and the perceived probability of default

changes to 10% per year.

Table 11.3 PV of ABC’s bond after a credit downgrade


1 5 10.00 0.900 00 2.5000 0.975 610 4.39022 5 10.00 0.810 00 2.6249 0.949 499 3.84553 105 10.00 0.729 00 2.7498 0.921 843 70.5625

Sum PV ¼ 78.7982

Yield ¼ 14.1562%


The promised cash flows do not change, but their expected value does. The new price is78.7982. The YTM on ABC bonds increases to 14.1562%, and the credit spread is now11.4145%.Our model can incorporate a deterministic scenario of changing credit quality.

Suppose we believe that, while the probability of default for year 1 is 0.11%, it willincrease to 6% in year 2 and 8% in year 3. We recomputed the cumulative survivalprobabilities, and the rest proceeds as before.

Table 11.4 PV of ABC’s bond assuming known credit deterioration


1 5 0.11 0.998 90 2.5000 0.975 610 4.87272 5 6.00 0.938 97 2.6249 0.949 499 4.45773 105 8.00 0.863 85 2.7498 0.921 843 83.6150

Sum PV ¼ 92.9454

Yield ¼ 7.7238%


Default Risk (Financial Math IV) and Credit Derivatives 299

The YTM on the bond changes to 4.9821%.In the last three examples, we showed how an estimate of default probabilities

translates into the credit spread through the calculation of the bond price as theexpected present value of the scheduled cash flows, where the expectation is weightedby the probabilities of receiving the cash flows. Next, we consider a model with morecomplicated changes to the credit quality of the issuer.

11.2 A CREDIT MIGRATION MODEL

The term ‘‘credit migration’’ refers to the possibility that the probability of default for agiven future period depends on the default risk of the bond during the previous period.That is, if we think that a bond is likely to be upgraded from A to AA or likely to bedowngraded from A to BBB over the next year, the probability of default for thefollowing year will depend on whether the bond was upgraded or downgraded. Inthis approach, we consider the probabilities not only of default, but more gradualchanges to the bond’s credit rating, and the path of those changes. We can also takeinto account the changing nature of the subsequent probabilities of upgrades anddowngrades.

Let us examine the following scenario. ABC has the probability of default over thenext year of 0.20%. We postulate that the annual default probability for year 2 willchange to 0.10% with a probability of 30% or to 0.70% with a probability of 70%. The30–70 probabilities of period-to-period changes in the issuer’s credit rating are referredto as transitional probabilities. Now, depending on where we end up in year 2, we willassume different transition probabilities for years 2–3 and different default probabilitiesfor year 3. If we end up at 0.10% for year 2, then we postulate that the annual defaultprobability for year 2 will change to 0.05% with a probability of 25% or to 0.30% witha probability of 75%. If we end up at 0.70% for year 2, then we postulate that theannual default probability for year 2 will change to 0.30% with a probability of 30% orto 3.00% with a probability of 70%. This way we can model a non-linear path towardcredit trouble and a migration from one credit category to the next over time.

We use a decision tree similar to the binomial tree for options to depict our scenario(ours will be recombinant, but it does not have to be). On each node, we mark theprobability of default for the year starting at that node and ending next year. We showtransitional probabilities of going up or down. We also show the scheduled cash flowfor the next period. The PV displayed at each node is the discounted expected value ofthe cash flow to be received next period, cumulated with the expected value of thesubsequent period’s cash flow which is probability-weighted and discounted. Graphic-ally, each node n is represented as:

Figure 11.1 A node of a credit migration tree.


where

PVn ¼ ð1�DefProbnÞ½CFnþ1 þ ðTransProbup � PVup þ TransProbdn � PVdnÞ�=ð1þ rn;nþ1ÞWe sweep backward through the tree to compute the expected value of all cash flowsunder our transitional and survival assumptions.

Figure 11.2 ABC’s bond in a credit migration model with transitional probabilities.

Today’s price of the ABC bond is 104.14 and the YTM is 3.5217%.Note that our tree does not rely on any hedging argument. It is just a convenient

graphical tool for computing weighted averages given conditional default and transitionprobabilities. It only coincidentally resembles option-pricing trees which are discreteapproximations of price dynamics.

11.3 ALTERNATIVE MODELS

Credit migration can, of course, be a lot more complicated and depend not just onprevious default probabilities, as in our model, but on other (state) variables. The lattercan be market-general (e.g., interest rates or economic indicators) or issuer-specific(e.g., financial ratios, legal structure). Any model has to be a vast statistical approxima-tion of the reality of defaults.One particularly well-known and highly qualitative model is that used by credit-

rating agencies. S&P’s and Moody’s assign issuers to credit quality categories withdifferent likelihoods of default. The agencies provide past default statistics. They alsopainstakingly analyze all relevant financial information to determine a forward-lookingforecast of an issuer’s ability to repay cash flows. This includes public financial dis-closure as well as general industry and economic data. We can refer to that as an expertopinion model.An interesting option-theoretic alternative is offered by a San Francisco-based

company called KMV. The approach is based on the observation that the stock ofany given company is a de facto call option on the total value of the firm with a strikeprice equal to the value of the debt. If we know the total value of the assets of the


company and the total value of the shares outstanding, then in effect we know the valueof the underlying price and the value of the option. If we can further assess the volatilityof the underlying assets, then, from the option-pricing model, we can back out thestrike price (the value of the debt) of the option (the equity). That, in turn, can beconverted to the YTM and the credit spread. It can also be converted to a forward-looking forecast of default probability. KMV claims to have superior out-of-samplepredictive power for the defaults. The enormous value added by the company’s servicelies in extremely thorough data collection process for all debt issues and a very rigoroustreatment. The approach relies on a hedging argument which, while quite impossible toimplement, does offer the implied self-consistent market consensus estimate of futuredefaults, in the same way as the implied volatility does for future price paths (here weare interested in the implied strike and its subcomponents though).

11.4 CREDIT EXPOSURE CALCULATIONS

FOR DERIVATIVES

Credit risk refers to the possibility that some cash flows that we have been promisedmay not be paid to us. In general, the magnitude of credit risk depends on two factors:the size of the cash flows owed to us and the default probability of the counterparty tothe transaction. A loan with twice the face value issued to the same borrower has twicethe credit risk. The same loan issued to two different borrowers may have differentcredit risks as the default probabilities of the two borrowers may be different. The priceof a credit-sensitive bond compensates the owner for both factors. Credit spreadimplied in the bond’s yield takes into account the probability of default. The yield isthen used in cash flow discounting: the greater the cash flows, the greater the PVreduction due to credit spread.

The notion of credit exposure refers to estimation of the size of the cash flows subjectto default (i.e., it only considers the first factor). In many cases, most notably inderivative contracts, cash flows are floating or contingent on market events and arenot known in advance. They have to be forecast. Forecasting is a very risky business.This renders credit exposure calculations a very inexact science. All that we can do is tryto come up with a metric of the relative riskiness of different assets based on the size ofthe exposed cash flows. Once that size is defined for a security, it can be cumulated withother securities of or contracts with the same counterparty to come up with the totalsize of cash flows exposed to the default of that counterparty. A bank performing sucha computation can then set a limit on the exposure it is willing to take. It can define acredit line limit for a given counterparty to regulate the business behavior of its tradersby inducing them to or preventing them from engaging in a particular type of deal inorder to manage the concentration of credit risks. From that perspective, it is a veryuseful, albeit inexact, business policy tool.

Credit exposure can be defined in many different ways. For example, credit exposureon a loan issued to a customer can be defined in future value terms as the interest andprincipal cash flows we have been promised on some future dates or as some kind of asingle PV concept. For bonds, credit exposure can be defined as the future cash flows


from the bond or their total PV. The latter definition will not lend itself to easyaggregation across other securities and contracts of the same issuer.When applying the concept of credit exposure to forwards, options, and other

derivatives, and depending on the structure of the transaction, the cash flows due tobe received may not be defined in advance. Consider a long position in an option. If theoption ends up in-the-money on the expiry date, then we are owed a cash flow; if it endsup out-of-the-money, then we are not owed a cash flow. Does this mean that if theoption is currently out-of-the-money we do not have any credit risk? No, we bought theoption, hoping that we might get a cash flow from it. There is a non-zero probability ofa positive cash flow if the option is not worthless. We do have credit exposure to theparty that sold us the option. If we used the simple logic of discounting cash flows usinga credit spread-corrected yield, as we can do with loans and bonds, all out-of-the-money options would have no computed credit exposure. So how else can we definecredit exposure?The answer is not simple. Whatever we do will have to be based on some probabilistic

assumptions about how prices or rates are going to move (i.e., our subjective estimateof market risk). It will have to rely on some guesswork. We will have to estimate theexpected value or a confidence interval value of the cash flows we are due. Whatever wedo will be just a guess or, more technically, a subjective estimate of some randomvariable. But, we get to choose what variable we want to estimate: the mean, a con-fidence interval cutoff, or something else.The most important principle in credit exposure calculations is consistency. We want

to treat assets and securities that are essentially the same equally. The credit exposureon a receive-fixed swap should not be any different from the net exposure on a longfixed coupon bond and a short floating-rate bond. The credit exposure on a forwardzero should not be any different from the credit exposure on two spot zeroes: one longand one short. We also want the credit exposure on otherwise identical contractsentered into with different counterparties to be identical. After all, the objective is tocompute the exposure first and then apply the credit rating of the counterparty todecide whether we are comfortable with it. Lastly, we have to be careful to definecredit limits consistently with the definition of exposure we adopt.Many forwards and derivatives are exchanges of cash flows, and we need to separate

the cash flows that we owe from the cash flows that we are owed. We do not have anycredit exposure on the first; we have credit exposure on the latter. But is our exposureequal to the full value of the cash flows that we are owed? Suppose we entered into a1-year foreign exchange (FX) forward to sell GBP 100,000,000 for USD 150,000,000.Our expected cash flows are an outflow of GBP 100,000,000 and an inflow of USD150,000,000. What if the counterparty to our FX forward defaults and on the maturitydate fails to deliver USD 150,000,000. We certainly are not going to rush to deliver ourpound sterling obligation under the contract. If the exchange rate moves against us andthe counterparty defaults, we will choose not to pay either; if the exchange rate movesin our favor, we will have lost a positive cash flow.One way to define credit exposure would be as a one-sided confidence interval cutoff

of the amount to be owed to us. Suppose today’s 1-year forward FX rate is USD1.50/GBP and the volatility (standard deviation) of the rate is 2%. Based on the bell curve,68% of the probability lies within �1 standard deviation, or �2%, of the central valuewhich we will assume is equal to the forward and 90% of the probability mass lies


within two standard deviations or �4%. The two standard deviation points arethus:

1:50ð1� :04Þ ¼ 1:44 and 1:50ð1þ 0:04Þ ¼ 1:56

At those points, the net cash flows in USD owed to us would be:

At 1.44 150,000,000� 100,000,000� 1:44 ¼ USD 6,000,000

At 1:56 150,000,000� 1,000,000� 1:56 ¼ USD � 6,000,000

We can define the 1.44 point as the 95% confidence interval cutoff. Based on ourestimate of volatility, there is a 5% chance that the rate would be at 1.44 and below.At 1.44 we are owed USD 6,000,000. We can define that as our credit exposure in futurevalue terms. We can also discount that amount to today to come up with the creditexposure in PV terms. Coincidentally, this would also be the 95% value-at-risk1

number. Note that if we did not use a confidence interval measure, but simplydefined the credit exposure in terms of its expected value, then our FX forward and,in fact, all on-market forwards would have zero credit exposures.

Let us provide another example. This time we look at an interest-rate swap. Supposewe have entered into a 5-year annual interest-rate swap to pay 5.5884% fixed against12% LIBOR (London interbank offered rate), on a notional principal of GBP100,000,000. Currently, the 12-month forward, spot zero, and par swap interest ratesfor different terms, stated in months from today, are as follows:

Table 11.5 Current rates

Forwards Maturity Zeros Maturity Swap

0� 12 5.000 000 12 5.000 000 12 5.000012� 24 5.500 000 24 5.249 703 24 5.243324� 36 5.761 423 36 5.420 001 36 5.406636� 48 6.000 000 48 5.564 702 48 5.542648� 60 5.800 000 60 5.611 720 60 5.5884

Our swap is on-market (i.e., it has a zero PV). How much money is at risk of default?Again we can try to determine a 95% confidence interval measure by consideringscenarios where money is owed to us at the five payment dates. We assume thatforwards are unbiased predictors of future LIBORs. Suppose the normal volatility ofLIBOR rates is 2% per year. This will have to be scaled up by the appropriate numberof years to set times. We will be owed money when LIBOR exceeds the fixed rate, sincewe are paying fixed. We consider two standard deviations up from the forwards.


1 Value-at-risk is a measure prescribed by the regulators, and used by the majority of banks, in OECD countries to describetheir market risk exposure. For a very readable treatment, see P. Jorion, Value at Risk (2nd edn), 2000, McGraw-Hill, NewYork.

Table 11.6 Credit exposure calculation using a two-standard-deviation interval

Time in months Forward Volatility 95% cutoff Cash flows Net CF Credit PV credit——————— LIBOR ———————————— exposure exposureSet Pay Fixed Floating

0 12 5.0000 0.00 5.00 (5,588,397) 5,000,000 (588,397)1 24 5.5000 2.00 5.72 (5,588,397) 5,720,000 131,603 131,603 118,8022 36 5.7614 4.00 6.22 (5,588,397) 6,222,336 633,940 633,940 541,1023 48 6.0000 6.00 6.72 (5,588,397) 6,720,000 1,131,603 1,131,603 911,2124 60 5.8000 8.00 6.73 (5,588,397) 6,728,000 1,139,603 1,139,603 867,347

Total 2,438,463

We compute LIBOR rates that correspond to the 95% interval. We compute the cashflows based on those projected LIBOR rates as well as fixed cash flows. We net the twoto get the amounts owed. If positive, we define those as our credit exposures. We alsocompute the credit exposure in PV terms. The interpretation of that PV number of£2,438,463 is that, based on predicted market movements and in particular the twostandard deviation movements, we stand to lose that amount in today’s pounds in caseof default. It should be said that discounting is not preferred here; it is better to knowcash flows at risk for all future dates.All of the above suggested methods are not based on solid science (i.e., the computed

amounts cannot be locked in, arbitraged, paid, or received). They should be viewedsimply as ‘‘risk scores’’ for relative exposure comparisons among products and counter-parties. They have no strict monetary interpretation. We cannot even be completelysure that a larger credit exposure number means greater credit risk due to a larger cashflow being exposed.

11.5 CREDIT DERIVATIVES

Credit derivatives, like other derivatives that transfer price or interest rate risk betweenparties, are structured finance products designed to transfer risk arising from a creditevent between two parties (e.g., they may provide exposure to or protection from thedefault of an issuer). Credit derivatives are forward-like products in that there is afuture date and the payout is defined as a net of two flows, at least one of which iscredit-related. More often, credit derivatives are option-like products in that there is atrigger event that can take place over a time period and the payout is contingent on theevent having taken place. If the event, which can be a default, restructuring, or a credit-rating downgrade, occurs, the credit derivative has a payoff. If the event does not occur,the credit derivative pays nothing.Similarly to standard options, the payoff of a credit derivative does not have to be

fixed, but it can depend on the ‘‘severity’’ or ‘‘size’’ of the event. The ‘‘severity’’ issimilar in concept to the notion of credit exposure (i.e., it measures the size of thecash flows lost due to default). This is closely related to the recovery value of thedefaulted security. The payoff can also depend on other market variables (prices andrates).Credit derivatives are very complex credit management tools offering lots of flex-

ibility, but often carrying additional market and credit risks. Their main appeal lies inallowing two parties to transfer the credit risk of a third party. This risk-sharingarrangement can be for a specific time period or can be demarcated by credit events.


Until credit derivatives started trading in the 1990s, the only credit risk managementtools at the disposal of bankers were a direct renegotiation with a borrower, diversifica-tion of the loan portfolio, or an outright sale of a loan. Credit derivatives allowed thecredit profile of a loan portfolio across many dimensions to be customized: length ofexposure, selling of some exposure but not all, correlating exposures by blendingexposures to like risks, relating the size of loan protection to other market variables, etc.

Basics

Let us consider a default protection option. The buyer of the option gets a payout if aspecified issuer defaults on or prior to some date. The buyer pays a premium as for anyoption or insurance product. The default is defined in terms of a reference security. Thedefault event is said to take place if the issuer of the reference security fails to paycoupon or any other cash flows specified by the reference security contract. More oftenthan not, the reference security is a bond. If the issuer of the bonds fails to pay on thebond, for the purposes of the default option the default has taken place and the optionhas a payoff. The payoff is then defined as some amount (say, par) minus the recoveryvalue. The recovery value is defined as the price of the defaulted reference security in theaftermarket. For example, if the bond falls in default, but trades at 60 in the after-market, the payoff on the option may be defined as 100 minus 60, or 40. The presump-tion is that the buyer of the protection owns the bond and can recover 60 by selling thebond, and is only compensated for the lost value.

Credit default swap

A default protection option is most often packaged as a credit default swap in which thepremium is paid periodically as opposed to all upfront. The term of the credit protec-tion and premium payment, called the maturity of the swap, is specified at the outset.As long as the premium continues to be paid, the protection is in place. This is similarto a term life insurance contract without any cancellation rights on both sides.

Let us illustrate. The Kool Kredit Bank has a portfolio of loans to different corpora-tions. Kool Kredit is worried about one particular loan to ABC Corp. which has fivemore years to maturity. ABC also has some bonds outstanding. Bonds and loans carrycross-default provisions. Kool Kredit selects one of these bonds as a reference securityfor the default swap it intends to enter into with a credit derivative dealer. In case ofABC’s default, the reference security is thought to stand to lose approximately the samepercentage of value as the Kool Kredit’s loan to ABC.

Figure 11.3 Credit default swap used as insurance against loan default (BRef is the value of thereference bond).

If ABC defaults, then the value of the loan to Kool Kredit will decline from par (100) to


the PV of any cash flows post-default. The value of the reference bond will also declineto the PV of any cash flows post-default (i.e., the recovery of the bond). Kool Kreditwill receive the difference between par and the recovery value. If Kool Kredit actuallyowned the reference bond and not the loan, then the credit derivative could have beenstructured even more simply with a physical-settle provision. Kool Kredit would havethe right to put the bond to the dealer in exchange for par. In the slightly more complexmark-to-market arrangement, the recovery value must be defined very clearly. Therecovery is typically defined as the bid or mid-price of the bond in the aftermarket.To avoid disputes, an additional provision may be inserted in the contract, specifyingthat five dealers may be polled to ascertain the bond’s aftermarket value.The International Swaps and Derivatives Association’s (ISDA) master documents

govern the language of credit derivative contracts. They specifically define the referenceentity (reference security), obligation (loan or bond provisions), credit event, and refer-ence obligation. Most trade confirmations include a materiality clause aimed atexcluding technical defaults with no change in the reference security’s value from thedefinition of a credit event.

Total-rate-of-return swap

Total-rate-of-return swaps allow banks to obtain default or value loss protection forloans on their books and a subsidy to their cost of funding in exchange for giving up thetotal return on a loan. The protection-buying bank essentially gives up any cash flowsfrom the loan without having to sell the loan or canceling the loan with the borrower.The protection-selling investor gains access to the return on the loan without having tofind his own customers, but bears the risk of default or any event leading to a loss ofvalue of the reference security and has to pay spread over LIBOR as compensation.Total return swaps are perfect examples of credit risk sharing where banks can diminishtheir exposure to the credit standings of certain borrowers, while non-bank investorscan gain exposure and return on loans to certain borrowers.As an illustration, consider Kool Kredit Bank: instead of buying into a credit default

swap, it enters into a total return swap with an investor or dealer. Kool Kredit does nothave to involve ABC in the transaction. It agrees to pay any interest and any apprecia-tion in the value of ABC’s loan to the dealer. The dealer agrees to pay Kool KreditLIBOR plus a spread and any depreciation in the value of the reference loan. In effect,the dealer’s compensation for providing credit protection is the difference between theinterest on ABC’s loan passed on to him and the spread over LIBOR he pays. The swapmay terminate on any payment exchange date or on a pre-specified credit event. At thatpoint, the two parties would enter into a settlement based on the mark-to-market (parminus the recovery value) of ABC’s loan which serves as the reference security for theswap.

Figure 11.4 Total-rate-of-return swap (BRef is the value of the reference bond).


The total return swap bundles credit protection with an exchange-of-interest-accrualformula. Credit protection is wide-ranging as the protection provider compensates forany loss of value on the reference security.

Credit-linked note

A credit-linked note is a debt security whose principal repayment depends on the valueof a third-party reference security. In case of third-party default, the defaulted securityis passed on to the investor in lieu of principal. The enhanced interest rate of the credit-linked note reflects the implicit sale of credit protection by the investor. The issuer ofthe note is typically a trust set up by a bank arranging the sale of the note. The trustuses the proceeds from the sale of the note to invest in risk-free securities to guaranteethe principal of the note if the third party does not default. If the third party defaults,the note is redeemed early at the recovery value of the reference security.

The investor gains a much enhanced interest rate, but bears all the risk of default.The arranging bank acquires protection for its loan or bond portfolio. In case of defaultof the underlying issuer, the bank receives the excess of the value of the securities heldby the trust over the recovery value of the reference security.

Figure 11.5 Credit-linked note (BRef is the value of the reference bond).

The credit-linked note bundles a credit default swap with a standard bond obligation.The credit protection premium paid by the arranging bank is passed on to the investorin the form of an enhanced coupon. The protection payout is passed from investors tothe bank at redemption. If there are no credit events associated with the referencesecurity, the investor gets a periodic coupon and par at maturity.

Credit spread options

Credit spread options offer payout dependent on the yield differential between default-risky securities and reference benchmarks. The payout is not contingent on any par-ticular credit event. Most credit spread options are structured as put options on theprice. The holder of a credit spread option has the right to sell a particular corporatebond at a contingent strike price. The strike price is computed to be the value thecorporate bond would have to have on the expiry date so that its YTM would beequal to the YTM on the reference, default-free security plus a spread strike. Forexample, the buyer of a spread option on ABC’s 5-year note struck at a spreadstrike of 50 bp over the 5-year U.S. Treasury note will face either of two possibilitieson the expiration date. If the spread between the YTM on the ABC bond and theTreasury is less than 50 bp, the option expires worthless. If the spread exceeds 50 bp, hesells the ABC bond to the option writer. The price he receives is equal to the PV of the


cash flows from the ABC bond discounted at the yield equal to the 5-year Treasuryyield plus 50 bp.Let us illustrate numerically. Suppose 1 year ago we purchased a 1-year European

option on the spread between ABC’s 6-year bond paying a 5% coupon semi-annuallyand the 5-year U.S. Treasury issue struck at 55 bp. The principal (face) amount was$100 million. At that time, the ABC bond yielded 5.75% (semi) to price at 96.2392. Thereference Treasury off which we were going to calculate the yield spread between thetwo bonds had not been issued. The then-current 5-year Treasury paying a 5.25% semi-annual coupon yielded 5.23% and priced at 100.1019. The option was described as 3 bpout-of-the-money since the spread at the time of the option purchase was 52 bp. Amonth ago, the U.S Treasury auctioned off a new 5-year note with a coupon of5.125%. This note will be used to determine the payoff on the spread option.Suppose the now 5-year ABC bond is yielding 5.68%. The current market price is

97.0759. The new 5-year Treasury is yielding 5.04% and prices to 100.3716. The yieldspread between the two bonds at 64 bp is higher than the strike spread of 55 bp. We willexercise our option to sell the bond at 55 bp over Treasuries (i.e., at a yield of 5.59%).Computing the price of the ABC bond at that yield we obtain 97.4571. Since we can buythe bond for 97.0759 on the open market, the payoff on the option to us is:

ð97:4571� 97:0759Þ=100 � $100,000,000 ¼ $381,153

Suppose instead that the now 5-year ABC bond is yielding 6.23%. The current marketprice is 94.7846. The new 5-year Treasury is yielding 5.59% and prices to 97.9958. Theyield spread between the two bonds (again at 64 bp) is higher than the strike spread of55 bp. We will exercise our option to sell the bond at 55 bp over Treasuries (i.e., at ayield of 6.14%). Computing the price of the ABC bond at that yield we obtain 95.1551.Since we can buy the bond for 94.7846 on the open market, the payoff on the option tous is:

ð95:1551� 94:7846Þ=100 � $100,000,000 ¼ $370,528

The option payout in the two scenarios is not identical, but almost the same. This isbecause it can be viewed as purely a bet on the spread between the two bonds grossedup by the duration of the bond under consideration. For the last scenario, the durationof the bond computed by blipping the yield from 6.14 to 6.15 is 4.1254. The payout onthe option is based on 9 bp (64� 55). So using the duration approximation, the totalpayout is:

0:041 254 � 9 � $100,000,000 ¼ $371,289

Note that nowhere in the calculations did we use the actual price of the Treasury. Bycomparing the two examples we can also see that the payout is independent of whetherinterest rates in general go up or down. The only relevant factor is the yield spreadbetween the chosen bond and the reference risk-free security. To emphasize that, spreadoptions are sometimes written by fixing the duration multiplier and defining the payoutonly with yields.The fundamental difference between credit spread options and other credit deriva-

tives is that credit events do not explicitly drive the payout, but are rather implicitlyincorporated in the payout through a spread differential. The greater the possibility ofdefault of the underlying bond, the greater the spread over risk-free assets is and thusthe greater the payout on the spread option is.


Credit options can also be written not on specific underlying bonds, but on theaverages of bond yields in a given credit category. These may be used by potentialfuture issuers to hedge against general spread increases till the time of issuance.

11.6 IMPLICIT CREDIT ARBITRAGE PLAYS

We briefly revisit some of the previously described arbitrage relationships to focus oncredit considerations. We take the perspective of a corporate debt issuer whose objec-tive is to obtain the lowest cost of financing in the capital market.

Credit arbitrage with swaps

Let us assume that a U.K.-based corporation has a choice of issuing debt in the U.K. orin Australia. Five-year swap rates are 5% in the U.K. and 7% in Australia. Given thecorporation’s credit rating, the debt can be issued at 60 bp over swap rates in the U.K.Suppose that credit perceptions are different in Australia and the corporation findsinvestors there willing to accept 45 bp over the swap rates. The corporation wouldprefer sterling-denominated debt as most of its operations are in the U.K., but thecost differential may push it to do the following.

The U.K. corporation issues a 5-year AUD-denominated bond with a coupon rate of7.45%. The bond sells at par. The corporation converts the proceeds into GBP tofinance its operations. It also enters into a fixed–fixed currency swap receiving 7.45%in AUD and paying 5% plus a margin in GBP. Given the current swap rates, themargin is likely to be a little over 45 bp, but much less than the 60 bp U.K. investorsdemand. The corporation benefits by obtaining financing at lower cost.

Callable bonds

A U.K.-based corporation currently trades at 80 bp over U.K. government gilts. Itbelieves that 10-year gilt rates will stay at the current 5% level or come down overthe next 3 years. It strongly believes that its own credit rating is likely to improvedramatically as it has several new products in the pipeline. The corporation issues10/2 callable bonds. The bonds have a 10-year final maturity, but can be called atpar starting 2 years from today. The YTM on the bonds is 5.95% which is higherthan a straight 10-year bullet yield of 5.80%. Suppose 2 years later the credit ratingof the corporation improves as predicted to 40 bp over gilts, but gilt rates have stayed at5%. The corporation may decide to call the bonds and issue new bullet notes at a 5.4%yield. Its overall cost of financing over the 10-year period will be lower than if it hadissued bullet bonds at the outset. In fact, the improved credit rating would have madebullet bonds more expensive to retire early as their price would have increased.

11.7 CORPORATE BOND TRADING

The objective of most banking operations, apart from earning fee income for services, isto acquire funds at the lowest possible cost and earning the highest possible rates on its


investments. The funds are obtained by offering checking, savings, and certificate ofdeposit (CD) accounts to retail customers or by borrowing in the wholesale marketsthrough overnight interbank or repurchase agreement (repo) markets. Investmentsinclude a variety of consumer and business loans. In this process, banks face twotypes of risks. The first is the duration mismatch; financing is likely to be short-term,while the investments are likely to be long-term. This then leads to reinvestment andliquidity risk. The second major risk is that of credit exposure to the default byborrowers.In order to diversify the credit exposure beyond a bank’s traditional customer base,

the bank may turn to the corporate bond market. This offers exposure to a widervariety of credits and potentially seniority over loans, but comes at the cost of alower yield.Corporate bond-trading operations generate two types of profit. The first is simply

from inventory turnover through market-making. By charging higher ask prices, payinglower bid prices, market makers pocket the spread. This is not without risk as they holdan inventory of bond positions: those bought for sale and those short-sold for repur-chase. This may at times be riskier than stock dealing as corporate bond markets areonly liquid for a limited number of better known issuers. The financing required formarket making is equal to the financing cost of net inventory plus any marginsrequired. The second type of profit from corporate bond trading comes from creditarbitrage. Credit arbitrage can be implicit or explicit. The first has to do with being netlong corporate bonds in the bank’s bond portfolio against the general financing fromthe bank’s treasury. Simply by buying bonds the bank is likely to make money in thesame way that the bank makes money on its loan portfolio, relative to the cheapfinancing from deposit taking and overnight operations. This is an easy way to generateprofit, but is subject to the same duration and liquidity risk as all the rest of the bankingoperation. The explicit type of credit arbitrage in corporate bond trading comes fromrelative value trades within the bond portfolio.Relative value arbitrage with corporate bonds can take on many forms. It can consist

of longs on one credit vs. shorts on another, if we believe that the first will improve andthe second will deteriorate. It can consist of longs or shorts on a credit vs. a position ina government security, if we believe that the credit spread over government securitieswill go down or up. It can also consist of sector and industry tilting of the portfolio, ifwe believe that the average spread on one industry is going to move differently relativeto that on another industry. Relative value arbitrage may also seek to exploit bondmaturity selection and reflect a view on the movement of the corporate spreads overtime, individually or in aggregate.In all these strategies, the important factor distinguishing outright speculation from

speculative relative arbitrage is whether the portfolio is hedged against the generalmovement of interest rates or not. This is done implicitly through duration matchingof the relative components of the strategy or explicitly by using government securities.Most corporate bond trading around the world is a cash business. Forwards on

corporate bonds are rare and options exist only in the embedded form. Theoretically,there is nothing stopping the markets to evolve to include both forwards and options.Curiously, this has not happened yet. Also surprising is the limited use of creditderivatives by corporate bond traders even though the reference securities underlyingmany of the derivatives come from the corporate bond markets. Likely, this has to do


with the skills gap between that required of a cash bond trader and that of a quanti-tative derivatives structurer. Also, most corporate bond trading is rarely linked to loantrading even though the two markets have to do with the same risk factors and theirintegration could lead to better credit diversification. Many banks obtain a bird’s-eyeview of the total credit exposure at the corporate level through some sort of ‘‘creditmetrics’’, but choose not to combine lending and bond trading as the infrastructure inthe two markets is different. This is clearly far from optimal.


_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________________________________________ Index _________________________________________________________________________________________________________________________________________________________

30/360 day-count convention 30–1, 34, 36, 113–16,

189–93, 204–5, 217–18

absolute basis, concepts 131–2

ABSs see asset-backed securities

Act/360 day-count convention 31, 68–74, 142,

168–70, 182–7, 266–7

Act/365 day-count convention 31, 83, 183–4, 217–18

Act/Act day-count convention 31, 36, 217–18

add-on/discount basis, zero-coupon rates 33

ADRs see American depositary receipts

AEY see annual equivalent yield

agencies, ratings 67, 77–8, 87–90, 124–5, 299–302

agency discount notes (discos), concepts 69, 85

agents, financial markets 1–2, 18

AIM 22

airlines 14

American depositary receipts (ADRs) 104–5, 107–8

American options

see also options

concepts 5–7, 236, 240–58

state-contingent claims examples 5–8

American Stock Exchange (AMEX) 18, 103, 107–8

amortization tables, concepts 38–9

amortizing, concepts 28, 38–40

annual equivalent yield (AEY) 32–3, 36

annual example, compounding 29–33

annuities, concepts 29–35

appendix 23, 297–312

APT models 125–6

arb desks see proprietary desks

arbitrage

see also dynamic . . .; pure . . .;

relative-value . . .; static . . .

cash-and-carry 155–6, 160, 175–98, 223–4, 275–9

concepts 2, 12–24, 45–9, 58–9, 85, 94, 109–11,

113–32, 150–9, 162–7, 175–98, 207–29, 240–2,

275–94, 310–12

credit derivatives 310

currency forwards and futures 156–9, 207–29

definitions 12–14

examples 12–16, 94, 113–32, 162–7, 207–29,

275–94

forward–forward arbitrage 175–6

forwards 156–9, 162–7, 175–98, 207–29

futures 150–9, 187–98, 207–29, 277–9

options 14–15, 240–2, 261, 275–94

program trades 108, 182–7

relationships 2, 162–7, 175–6, 207–29, 311–12

speculation contrasts 14–16, 21–2, 294, 311–12

spot markets 113–32, 162–7, 182–7

spot–forward arbitrage 162–7, 175–98

stock index futures 150–4, 175–6, 182–6, 223–4,

277–9

stock indexes 150–4, 175–6, 182–7, 223–4, 277–9,

285

swaps 94, 207–29, 310

term structure of interest rates 45–9

triangular arbitrage 15, 110–11, 129–30

volatilities 279–94

warrants 278–9

Argentina 111

asset transformers

see also banks; financial institutions; insurance

companies; mutual funds

concepts 16–18

asset-backed bonds 9

asset-backed securities (ABSs) 9, 24, 67, 75–6,

96–9

see also fixed-income . . .

concepts 96–9

international comparisons 97–9

prepayment risks 97, 222–3

spreads 97–8

statistics 97–9

tranches 97

types 97–8

assets, convenience/pure assets 160–2, 173

assignments, swaps 90

at-the-money options

see also options

concepts 236, 261–4, 267, 283

auctions, bonds 84, 86

AUD 64–6

Australia 64–6, 75–6, 87, 109–11

auto loan ABSs 97–8

backwardation concepts 160–2

balloon loans/repayments, concepts 38

Bank of England 86–7

Bank for International Settlements (BIS) 67, 74,

80–3, 225–8

Bankers’ acceptances (BAs), concepts 74

banks

see also commercial . . .; investment . . .

commitment fees 200

concepts 1–2, 16–18, 56, 146–9, 163–5, 199–200,

272, 310–12

corporate bonds 310–12

currencies 109–11

overnight refinancing 70

barbell–bullet combinations, spot

relative value trades 116–17

Barings 16

BARRA 125–8

barrier options

see also options

concepts 256, 272–4

BAs see Bankers’ acceptances

basis points, price values 58, 197

basis risks, concepts 294

basis trades, concepts 14, 113, 131–2, 188–9

BBA see British Bankers Association

Belgium 71, 77, 104

benchmark bonds 13, 83–4, 91, 141–9, 175–6, 193–8

benchmarks 13, 24, 83–4, 91, 125–9, 141–2, 175–6,

193–8, 287

Bermudan options

see also options

concepts 269–72

best-of options, concepts 288

betas 60, 125–9

BEY see bond equivalent yield

bid–ask spreads

concepts 47, 68–9, 110, 129–30, 211

currencies 110, 129–30

binary bets, concepts 235

binary options

see also options

concepts 240, 246–7, 256, 272–4

binomial pricing, options 244–64, 276–94

BIS Quarterly Review 225–8

BIS see Bank for International Settlements

Black, Fischer 256–64

Black–Scholes model 256–64, 279–85

blipping method, concepts 195–9

Bloomberg 89, 154–5, 188, 228

bond equivalent yield (BEY) 32–3

bond exchanges, historical background 18

bond-equivalent basis, YTMs 36

bonds 1, 6, 8, 11–15, 18–19, 23–4, 27–58,

110–11, 141–9, 187–92, 200, 265–9,

278–9, 283–9, 290–2

see also corporate . . .; coupon; fixed income

securities; government . . .

auctions 84, 86

benchmark bonds 13, 83–4, 91, 141–9, 175–6,

193–8

callable bonds 11–13, 85, 221–2, 289–92, 310

cheapest-to-deliver bonds 188–9

concepts 1, 6, 8, 11–13, 15, 18–19, 23–4, 27–58,

79–90, 110–11, 141–9, 187–92, 200, 265–9,

278–9, 283–9, 290–2

convertible bonds 6, 11, 59, 289–90

credit covenants 11

credit derivatives 19–20, 297–312

credit risks 15, 20, 23–4, 113, 137–41, 147, 276,

295–312

day-count conventions 31–6, 83, 113–16, 142,

168–70, 189–92, 204–5, 217–18, 266–7

dual-currency bonds 290–1

embedded options 58, 173–4, 228, 289–92, 311–12

emerging markets 79–83, 97

financial mathematics 27–58, 128–9

floating rate bonds 39–40, 51, 54, 92–4, 165–7,

170–3, 200, 203–7, 209–10, 219–21, 268–9

flow trading 110, 121–3

forwards 141–9, 187–92

futures 141–9, 187–92

hedging 8, 13

historical background 18–19, 33–4


issuers 79–90, 200–5

maturities 40–9, 88, 94, 113–16, 167–70

options 265–9, 278–9, 283–92

pass-through certificates 95–6, 292

price/yield relationships 49–58, 148

puttable bonds 291–2

range bonds 222

ratings 40–1, 59, 77–8, 87–90, 124–5, 299–302

registration requirements 34

spot relative value trades 113–25

spot–forward arbitrage 187–92

statistics 79–90

structured finance 218–23, 289–92

subordinated bonds 87–8

valuations 49–58, 170–3

warrants 278–9, 291

zero-coupon rates 27–8, 32–49, 167–70

book-running activities

options 279–83

swaps 210–17

bootstrapping, concepts 44–9, 167–70, 189–93,

286–7

borrowing-against-the-box strategies, options 275–6

borrowing/lending rates

futures 143–9, 175–6, 181–7, 190–8

options 256, 275–6

brands 59

Brazil 77, 111, 162

Brealey, Richard A. 60, 125

314 Index

British Bankers Association (BBA) 147

broker-dealers

see also financial institutions

book-running activities 210–17, 279–83

concepts 2, 16–18, 21–2, 71–2, 102–3, 121–3,

210–17, 225–9, 233–4, 279–94

purposes 2, 16, 21–2, 71–2, 121–3, 210–17, 279–94

swaps 210–17, 225–9

bullet bonds

see also corporate bonds

concepts 6, 12–13, 85, 87–90, 116–17, 290–2

valuations 6

bullet–barbell combinations, spot


bundling concepts, information 11–12

bunds

see also government bonds

concepts 86–7, 141, 217

businesses 9, 18–20, 59

see also companies

equity/debt needs 9, 18–20

historical background 18

buy-side participants

see also speculation

concepts 21–2

buy–write strategies, options 238–40

buyback programs 102, 109

CAC-40 106, 149–50

calendar spreads, concepts 238–40

call options 5–8, 235–94

see also options

concepts 235–94

call spreads, concepts 237–40

callable bonds 11–13, 85, 221–2, 289–92, 310

Canada 75–6, 109

capital asset pricing model (CAPM) 60–4, 125

capital flows

concepts 1–24, 108–9, 234–5

financial markets 1–24, 108–9, 234–5

opportunity costs 20–1, 27

capital markets

see also bonds; equities


capitalization-weighted indexes, concepts 106–7

caplets, concepts 244, 265–9, 286–8

CAPM see capital asset pricing model

capped floaters, concepts 221

caps 10–11, 219–20, 244, 264–9, 273, 286–8, 291–2

car lease agreements 11

cash claims, delivery types 2

cash flow discounting

concepts 23–4, 27–66, 101–12, 119–21

equities 59, 101–9

interest rates 27–8, 32–40, 42–58, 119–21

cash markets see spot markets

cash-and-carry

arbitrage 155–6, 160, 175–98, 223–4, 275–9

concepts 14–15, 23–4, 135–6, 155–6, 160, 175–98,

223–4, 275–9

options 275–9

reverse cash-and-carry 135–6, 152, 160, 187

spot–forward arbitrage 155–6, 160, 175–98

cash-for-paper exchanges 9

cattle markets 111–12

CBOs see collateralized bond obligations

CBT see Chicago Board of Trade

CDDs see cooling degree days

CDOs see collateralized debt obligations

CDs see certificates of deposit

certainty equivalence

concepts 146–7, 149, 167, 170

Eurodollars 146–7, 167

forward rate agreements 149, 167

futures 146–7, 170

certificates of deposit (CDs) 16–17, 20–1, 29–33, 37,

56, 68, 74, 311

cheap trades 115

cheapest-to-deliver bonds (CTD), concepts 188–9

chessboards, state-contingent claims 3–8, 193

CHF 64–6, 130, 262–4

Chicago Board of Trade (CBT) 137–41, 187–92

Chicago Mercantile Exchange (CME) 136, 141–2,

149, 158–9, 168, 212

China 104

CHIPS 73, 109–10

CIRP see covered interest-rate parity

Citibank 92

claims

see also state-contingent claims

concepts 1–8

financial markets 1–8

methods 1–2

clean prices, concepts 36

clearinghouses, futures 137–41, 147

CLNs see credit-linked notes

CLOs see collateralized loan obligations

CME see Chicago Mercantile Exchange

CMOs see collateralized mortgage securities

CMTs see constant maturity options

Coca-Cola 92

Cohen, B.H. 89

collateral

asset-backed securities 96–9

debentures 87

mortgage-backed securities 94–6

repurchase agreements 71–2, 85

collateralized bond obligations (CBOs), concepts

98–9

collateralized debt obligations (CDOs), concepts

98–9

Index 315

collateralized loan obligations (CLOs), concepts

98–9

collateralized mortgage securities (CMOs), concepts

95–6

commercial banks

concepts 1, 16–17, 56, 70

Fed Funds 70, 286


portfolio immunization 56–7

commercial paper (CP) 9, 70, 74–9

see also money markets

concepts 74–9


issuers 75, 77

commitment fees, banks 200

commodities 1, 18, 64, 101, 111–12, 131–2, 135–6,

138–41, 160–2, 173–4, 224–9, 275–6, 277–8,

288, 293–4

basis trades 131–2

concepts 111–12, 131–2, 138–41, 160–2, 173–4,

224–9, 275–8, 293–4

electricity markets 112, 173–4, 294

futures 111–12, 131–2, 135–6, 138–41, 160–2,

173–4

long-dated commodity options 293–4

options 275–6, 277–8, 288, 293–4

over-the-counter markets 111–12, 173–4, 224–9


statistics 225–9

swaps 224–9

types 111–12, 138–41, 161–2

commodity exchanges, historical background 18

companies

see also businesses; corporate bonds

debt values 59


liquidations 59, 298–300

objectives 18, 59

valuations 59

compensation

see also returns

risk 2–8

composite payoffs, options 236–40

compounding

concepts 29–40, 50–1

examples 29–33

constant maturity options (CMTs) 273, 287–8

contangos, concepts 160

contingent delivery times

see also delivery times; options


convertible bonds 11

examples 10–12

continuous rebalancing, concepts 176, 211, 262–4,

279, 289

contract sizes, futures 140–1

convenience assets, concepts 160–2, 173

conversion factors, concepts 187–8

convertible bonds

see also hybrid securities

concepts 6, 11, 59, 289–90

contingent delivery times 11

valuations 289–90

convexity

concepts 49–58, 116–25, 148, 213, 273

definition 57

duration 55, 57, 116–25

interest-rate risk 51, 55, 57, 116–25

negative convexity 118–21


cooling degree days (CDDs) 137

corporate bonds

see also bonds; bullet bonds

banks 310–12

categories 87–8

concepts 8, 19, 34, 79–83, 87–90, 121–3, 310–12

coupon 34, 121–3

credit derivatives 311–12

day-count conventions 31, 204–5, 217–18

Eurobonds 33–4, 88

forwards 311–12

hedging 8

international comparisons 79–83, 87–90

investment-grade bonds 87–8, 124–5

maturities 88

non-investment-grade bonds 87–8, 124–5

options 311–12


secondary markets 87–90


spread strategies 121–3

statistics 79–83, 87–90

subordinated bonds 87–8

yield curve trades 123–4

corporate finance see investment banks

correlation risks

see also tracking risks

concepts 186–7, 288, 294

cost-of-capital concepts 20–1

cost-of-carry arguments, concepts 23–4,

135–8, 153–6, 174, 251

counterparty default, swaps 94, 306–10

coupon

see also bonds

bootstrapping 44–9, 167–70, 189–93

concepts 27–8, 32–58, 83–5, 87, 113–16, 162–7,

191–3, 199–229

deferred coupons 87

frequency issues 31–6, 51, 83–5

historical background 33–4

replication strategies 113–16, 162–7

covered calls 7, 238–40

316 Index

covered interest rate parity (CIRP), currency

forwards 156–8, 176–87, 208–10

Cox, John C. 244

credit card ABSs 97–8

credit covenants, bonds 11

credit default swaps, concepts 306–10

credit derivatives 19–20, 297, 305–12

arbitrage 310

calculations 305–12

concepts 19–20, 297, 305–12


examples 305–12


payoffs 305–12

types 305–10

credit migration, concepts 300–1

credit ratings see ratings

credit risks 15, 20, 23–4, 94, 113, 137–41, 147, 276,

295–312

concepts 15, 20, 23–4, 113, 137–41, 147, 276,

295–312

credit migration 300–1

credit spreads 8, 56, 162–7, 297–312

default probabilities 94, 298–300

financial mathematics 297–312

forecasts 302–5

standard deviations 303–5

credit spreads

concepts 8, 56, 162–7, 297–312

options 308–10

credit-linked notes (CLNs) 308

CRISP 128

cross-currency swaps

see also swaps

concepts 92–3, 110

cross-ratios, currencies 64, 110, 158–9

crossing networks 103

CTD see cheapest-to-deliver bonds

Cubes 18

currencies 1, 12, 14–15, 19, 23–4, 64–6, 72–4, 101,

109–11, 129–30, 154–9, 176–87, 200–29, 242–3,

277–8, 283–9, 290–1, 303–5

see also individual currencies

analysts’ mean forecasts 66

banks 109–11

bid–ask spreads 110, 129–30

concepts 64–6, 72–4, 101, 109–11, 129–30, 154–9,

176–87, 242–3, 277–8, 283–9, 290–1, 303–5

covered interest rate parity 156–8, 176–81, 208–10

cross-currency swaps 92–3, 110

cross-ratios 64, 110, 158–9

dual-currency bonds 290–1

emerging markets 111

Eurocurrencies 72–4, 79, 141–9, 166–7, 175,

191–8, 200, 211–29, 264–9

exchange controls 110–11


fixed-for-fixed currency swaps 200–3,

205–7, 208–10

forwards 109–10, 154–9, 176–87, 207–29, 303–5

futures 154, 158–9, 182–7, 207–29


market segments 109–10

one-price law 66

options 242–3, 261–2, 275–8, 283–9, 290–1

over-the-counter markets 109–10, 225–9

quote conventions 64–5, 109–10, 154–5, 242–3

spot relative value trades 129–30, 176–81


swaps 92–3, 110, 200–29, 267–72

transaction costs 66, 129–30

curve-blipping method, concepts 195–9

custom baskets 107–8, 153–4

databases, returns 128–9

DAX 106, 149–50, 182

day-count conventions, concepts 30–6, 68–74,

113–16, 142, 168–70, 182–7, 189–92, 204–5,

217–18, 266–7

DDM see dividend discount model

dealers

book-running activities 210–17, 279–83

broker-dealers 2, 16–18, 71–2, 102–3, 121–3,

210–17, 225–9, 233–4, 279–94

debentures 9, 34, 87–8

concepts 87–8

debt 9, 18–19, 59, 199–229

business needs 9, 18–19

ratings 40–1, 59, 67, 77–8, 87–90, 124–5, 200,

299–302

values 59

debt securities see fixed income securities

default protection options, concepts 305–6

default risks 15, 20, 23–4, 94, 113, 137–41, 147, 276,

295–312

see also credit risks

deferred coupons, concept 87–8

delivery times, concepts 9–12, 23–4

delta hedging, concepts 12–13, 23–4, 233–4, 245–58,

279–94

depositary receipts (DRs) 104–5, 107–8

derivatives 67, 225–9, 297–312

see also forwards; futures; options; swaps


statistics 225–9

Deutsche Borse 19, 104

developing countries, secondary markets

19–20

digital bets, concepts 235

digital options

see also options

concepts 273–4

Index 317

dirty prices, concepts 36

discos see agency discount notes

discount basis/add-on, zero-coupon rates 33

discount yield, definition 68

discounting see cash flow discounting

disintermediation, concepts 18

distressed debt 97–9

diversification strategies, concepts 9–10, 17, 126–8,

293

dividend discount model (DDM)

concepts 60–4

examples 60–3

dividends

see also equities

concepts 59–64, 152–4, 183–7, 257–8

options 257–8

retained earnings 63–4

stock index futures 152–4, 182–7, 223–4

domestic markets, diversification strategies

9–10

Dow Jones 106, 186–7

downward-sloping yield curves 42, 118–21

Drexel Burnham Lambert 20

DRs see depositary receipts

dual-currency bonds, concepts 290–1

duration

see also modified . . .

computer calculations 55

concepts 51–7, 116–21, 197–8

convexity 55, 57, 116–25

definition 52

dynamic duration hedging 197–8

examples 52–6

floating rate bonds 54

guesses 53–4

interest-rate risk 51–6

linear approximations 54–5

portfolio duration 56–8

zero-coupon bonds 54

duration matching

see also portfolio immunization

concepts 13, 56–7, 116–21, 176


Dutch auctions, bonds 84, 86

dynamic arbitrage, concepts 12–16, 116–21, 176,

193–8, 279–94

dynamic duration hedging, concepts 197–8

dynamic hedging, concepts 193–8, 211, 251–8

dynamic relative arbitrage, concepts 12–16, 116–21,

176, 193–8, 279–94

EAR see equivalent annual rate

East Asia, debt burdens 79

eBay 154, 186

EBS 109

ECNs see electronic communications networks

economic issues

options 294

secondary markets 108–9, 234

term structure of interest rates 42

EDs see Eurodollars

efficiency issues, investment/saving flows 1–2, 18–19,

108–9, 234–5

electricity markets 112, 173–4, 294

electronic communications networks (ECNs) 103,

105

embedded options 58, 173–4, 228, 289–92, 311–12

concepts 228, 289–92, 311–12

option-adjusted spreads 96, 291–2

emerging markets

asset-backed securities 97, 98–9

bonds 79–83, 97, 98–9

currencies 111

energy markets 112, 173–4, 294

Enron 20, 174

equities 3–9, 18–20, 23–4, 58–64, 67, 101–9, 125–9,

223–9, 235–79

see also dividends; returns

business needs 9, 18–20

buyback programs 102, 109

cash flow discounting 59, 101–9

concepts 3–4, 9, 18–20, 58–64, 67, 101–9, 125–9,

223–9


depositary receipts 104–5, 107–8

dividend discount model 60–4


growth companies 63–4



market model 125–8

market orders 105–6

new issues 19, 101–2, 108–9

options 235–94

pairs trading 14

portfolio strategies 125–9

price rises/falls 9

price/earnings ratios 63–4

primary markets 18–20, 101–2, 108–9


relative value trades 125–9, 182–7



statistics 101–9, 128–9, 225–9

stock indexes 15, 106–7, 125–9, 137–8, 149–54,

175–6, 182–7, 223–4, 243–4, 262–4, 277–9, 285

swaps 223–9

underwriting practices 101–2

valuations 60–4, 101–9

equivalent annual rate (EAR), concepts 30–3

ESCB 70

ETFs see exchange-traded funds

318 Index

EUR 64–6, 71–2, 109–11, 130, 154–9, 176–81, 212,

227–8, 262–4, 273–4

Eurex 137, 141, 149–50, 174

Eurobonds 33–4, 88

Eurocurrencies 72–4, 79, 141–9, 166–7, 175, 191–8,

200, 211–29, 264–9

see also money markets; swaps

concepts 72–4, 79, 141–9, 166–7, 175,

191–8, 200, 211–12

futures 141–9, 166–7, 175, 191–8, 211–12

options 264–9

TED spread 74

Eurodollars (EDs) 31, 72–4, 79, 141–9, 167–70,

191–8, 212–29

Euronext 104

Europe 17–18, 32, 36, 70–2, 75–90, 103–4, 107, 130,

154–9

see also individual countries


bond markets 79–90

commercial paper 75–7

corporate bonds 79–83, 87–90

currencies 64–6, 71–2, 109–11, 130, 154–9,

176–81, 212, 227–8, 262–4, 273–4

day-count conventions 32, 36, 217–18

electricity markets 173–4

equity secondary markets 103–4

financial institutions 17–18

government bonds 79–83, 86–7


repurchase agreements 71–2

European options

see also options


exchange controls, currencies 110–11

exchange-traded funds (ETFs) 18, 107, 153–4

exchanges, concepts 2, 18–19, 234–5

exotic options

see also options

concepts 272–4

expiry considerations, options 235–41, 244–64,

276–94

exposures 9–10, 15, 20, 23–4, 113, 137–41, 147, 276,

295–312

factor rotation, concepts 128

factor-based models

concepts 125–8

returns 125–8

fair values, concepts 135–6, 150–4, 155–6, 160,

177–87

Fama–French model 126–7

Fannie Mae see Federal National Mortgage

Association

Farm Credit Bank 69

Fed Funds (FFs), U.S. 70, 92, 286

federal agency discount notes see agency discount

notes

Federal Home Loan Bank 69

Federal Home Loan Mortgage Corporation

(Freddie Mac) 69, 85

Federal National Mortgage Association (Fannie

Mae) 69, 85

Federal Reserve, U.S. 67–70, 261–2

Fedwire 73, 109–10

FFs see Fed Funds

financial contracts


concepts 2–3

financial institutions 2, 8, 11–12, 15–18, 22, 70,

94–6, 225–9

see also asset transformers; broker-dealers

categories 16–18

concepts 2, 8, 11–12, 15–18, 22, 70, 94–6, 225–9

international comparisons 17–18, 22

U.S. 17–18, 70

financial markets

agents 1–2, 18

claims 1–2


delivery times 9–12, 23–4

economic descriptions 9

fundamental issues 1–24, 234–5

motor car analogies 1, 22–3

overview 1–24

purposes 1–24, 234–5

segments 1–2

structure 1–2, 8–12

time considerations 9–12

financial mathematics 15, 23–4, 27–66, 128–9

bonds 27–58, 128–9


currencies 64–6

equities 58–64

forwards 135–74

futures 135–74

options 231–74

spot 27–66, 128–9

Financial Services Authority (FSA) 106

Financial Times World Indexes 106

FINEX 158–9

Finland 77, 98

fixed coupon stripping, bonds 15, 166

fixed-for-fixed currency swaps, concepts 200–3,

205–7, 208–10

fixed-for-floating interest-rate swaps, concepts

203–7, 209–10, 222–3

fixed income securities 1, 6, 9, 19–20, 24, 27–58,

67–99, 113–25, 187–92, 200–5, 297–312

see also asset-backed . . .; bonds; money markets;

mortgage-backed . . .; swaps

concepts 67–99, 113–25, 187–92, 200–5

Index 319


credit risks 15, 20, 23–4, 113, 137–41, 147, 276,

295–312

day-count conventions 30–1, 83, 113–16, 168–70,

189–92, 204–5, 217–18, 266–7

interest rates 9, 113–25, 200–5

spot relative value trades 113–32, 187–92



flat yield curves 42

floating-for-floating swaps

see also swaps

concepts 92–3

floating rate bonds

concepts 39–40, 51, 54, 92–4, 165–7, 170–3, 200,

203–7, 209–10, 219–21, 268–9

duration 54

interest-rate risk 51

valuations 170–3

floorlets, concepts 265–9, 286–8

floors, interest rates 11, 264–9, 273, 286–8

flow trading

bonds 110, 121–3


forecasts, credit risks 302–5

foreign exchange (FX) see currencies

foreign markets, diversification strategies 9–10

forward discounts/premiums, concepts 155

forward rate agreements (FRAs)

certainty equivalence 149, 167

concepts 147–9, 163–4, 167–70, 175–6, 180–1,

189–92, 207–29, 267–72

futures convexity 148–9

mechanics 148–9, 163–4

options 267–72

statistics 225–9

forward–forward arbitrage, concepts 175–6

forward–forward volatility, concepts 286–7

forwards 2, 4–5, 6, 10–11, 23–4, 109–10,

133–229, 241–2, 251, 267–9, 303–5, 311

arbitrage 156–9, 175–98, 207–29

borrowing/lending rates 143–9, 175–6, 181–7,

190–8

cash-and-carry 14–15, 23–4, 135–6, 155–6, 160,

175–6, 223–4

concepts 2, 4–5, 6, 10–11, 23–4, 109–10, 133–229,

241–2, 251, 267–9, 303–5, 311

convenience assets 160–2, 173



currencies 109–10, 154–9, 176–87, 207–29, 303–5

definition 2, 135

energy markets 173–4

examples 137–8

fair value concepts 135–6, 155–6, 160, 177–87


futures contrasts 135–8

information 12

interest rates 141–9, 156–8, 162–7

obligations 4–5

options 241–2, 251, 267–72, 275–9

pure arbitrage 12–13, 175–6

repurchase agreements 171–3, 187–92

risk-sharing activities 137–8

spot contrasts 10



swaps 199, 207–29, 270–2

synthetic forwards 163–7, 175–98, 241–2, 275–6

synthetic zeros 164–5, 189–98

U.S. 136–74, 176–98

valuations 12, 135–6, 155–6, 160, 177–87

yield curves 167–73, 189–98, 211–29

zero curves 167–70, 189–98, 211–29

zero-coupon rates 164–70, 189–98, 211–29

France 71, 76, 86–7, 98, 104, 106, 217

FRAs see forward rate agreements

Freddie Mac see Federal Home Loan Mortgage

Corporation

FSA see Financial Services Authority

FTSE 100 104, 106, 149–50, 274

future value interest factors, concepts 28–32

futures 10–11, 14–15, 23–4, 106, 111–12, 131–2,

135–74, 182–98, 223–4, 277–9

arbitrage 150–9, 187–98, 207–29, 277–9


bonds 141–9, 187–92

borrowing/lending rates 143–9, 175–6, 181–7,

190–8

clearinghouses 137–41, 147

commodities 111–12, 131–2, 135–6, 138–41,

160–2, 173–4

concepts 10–11, 14, 23–4, 111–12, 135–74,

182–98, 223–4

contract sizes 140–1

convenience assets 160–2, 173

currencies 154, 158–9, 182–7, 207–29

definition 135

energy markets 173–4

Eurocurrencies 141–9, 166–7, 175, 191–8, 211–29

examples 136–74

fair value concepts 135–6, 150–4, 160,

177–87


forwards contrasts 135–8

gambling elements 136


interest rates 141–9, 156–8, 162–7

mechanics 135–74

multiplier concepts 139–49, 159, 168–70

pricing theory 150–4

320 Index

fixed income securities (cont.)

risk-sharing activities 137–8

single stocks 153–4


stock index futures 15, 137–8, 149–54, 175–6,

182–7, 223–4, 262–4, 277–9

temperature readings 136–7

trading floors 137–8

types 136–74

U.K. 141–9, 153–4

U.S. 136–74, 187–92

valuations 135–6, 150–4, 160, 177–87

variation margins 135–41, 150–4

futures convexity, forward rate agreements 148–9

FX see currencies

gambling

dangers 16

futures 136

options 233–5, 288, 294

gammas, concepts 260

GBP 64–6, 71, 93, 109–11, 200–29, 262–4, 303–5,

310

GDP, options 294

GE Capital 75

general obligation bonds (GOs) 89

geographical locations, hedging 22

Germany 20, 71, 76, 86–7, 98, 104, 106, 109–10, 141,

217, 228

gilts

see also government bonds

concepts 86–7

Ginnie Mae see Government National Mortgage

Association

Glass–Steagal Act 17–18

GlaxoSmithKline (GSK) 14

global economy 1

global investments, savings 1

GMAC 75

GOs see general obligation bonds

government bonds

concepts 1, 8, 19, 31, 34, 79–94, 113–21, 211,

297–8, 310

coupon 34

day-count conventions 31, 83, 113–16, 217–18


nicknames 86–7

risk-free government bonds 8, 297–8



statistics 79–90

swaps 90–4, 211, 217–18

twos–tens trades 117–21

Government National Mortgage Association

(Ginnie Mae) 69

grain markets 111–12

greed 15

Greenspan, Alan 69

growth companies, concepts 63–4

GSK see GlaxoSmithKline

hazard insurance 10–11

see also insurance . . .

HDDs see heating degree days

heating degree days (HDDs) 136–7

hedge funds

concepts 22

market-neutral hedges 125–9

hedge ratios

basis trades 131–2

options 245–64

hedging 2, 8, 13–16, 20–2, 24, 84, 119–21, 125–9,

160–2, 193–8, 211, 233–4, 245–64, 279–94


blipping method 195–9

commodity futures 161–2

concepts 2, 8, 13–16, 20–2, 24, 84, 119–21, 125–9,

160–2, 193–8, 211, 233–4, 245–64, 279–94

corporate bonds 8

definition 21

delta hedging 12–13, 23–4, 233–4, 245–58, 279–94

dynamic hedging 193–8, 211, 251–8

examples 8, 13–16, 20–2

geographical locations 22

insurance companies 293

interest rates 8, 119–21

multi-dimensional cubes 8

options 245–64, 279–94

relative value arbitrage 12–16, 21–2

risk-sharing activities 8, 21–2

speculation contrasts 20–2, 24, 311–12


swaps 211–29

high-yield bonds see non-investment-grade bonds

historical volatility, concepts 260–1, 293

HKD 64–6, 104

holding company depositary receipts (HOLDRs) 18

holding period return (HPR)

concepts 35–8

examples 36–7

HOLDRS 107–8

home equity loans 96–7

HPR see holding period return

Hull, John C. 257

hybrid securities

see also convertible bonds

concepts 59

IG Metallgesellschaft 16

illiquid markets, bid–ask spreads 47

implied volatility

concepts 13, 260–4, 279–94

realized volatility 262

smiles 261–4, 282–3

Index 321

in-the-money options

see also options

concepts 236, 241, 259–60, 263–4, 283

index principal swaps (IPSs), concepts 222–3

index-linked gilts, U.K. 86–7

indexes

see also stock indexes

futures 15, 106, 137–8, 149–54, 175–6, 182–7,

223–4, 262–4, 277–9

options 10–11

India 77

inflation

index-linked gilts 86–7

options 294

Treasury inflation-protected securities 85

information

bundling concepts 11–12

forwards 12

options 12

secondary markets 19

spot markets 11–12

valuations 11–12

initial public offerings (IPOs) 19, 102, 108–9, 242

innovations 224–5, 293

institutional traders, concepts 8, 11–12, 15–18

insurance, options 233–4, 241, 278–9, 292–4

insurance companies 1–3, 6, 17, 21–2, 241, 278–9,

293–4

concepts 1–2, 17, 21–2, 293–4

hedging 293

insurance contracts 3, 6, 10–11, 233–5, 241, 278–9,

292–4

contingent delivery times 10–11

examples 3, 6, 234–5, 241, 292–4

types 10–11

interest payments

see also coupon

concepts 1, 24, 27–8, 67, 88–9, 94–6

time value of money 27

interest-rate risk

concepts 49–58

convexity 51, 55, 57, 116–21

duration 51–6

elimination methods 56

floating rate bonds 51

zero-coupon rates 51–2

interest rates

basics 27–32

caps 10–11, 219–20, 244, 264–9, 273, 286–8, 291–2

cash flow discounting 27–8, 32–40, 42–58, 119–21

compounding 29–33

concepts 10–11, 27–66, 89–94, 119–21, 141–9,

156–8, 162–7, 219–20, 225–9, 244–69, 273,

286–8

covered interest rate parity 156–8, 176–87, 208–10

day-count conventions 30–6, 68–9, 113–16, 142,

168–70, 182–7, 189–92, 204–5, 217–18, 266–7

duration 51–6, 116–21

fixed-for-floating interest-rate swaps 203–7,

209–10, 222–3

fixed income securities 9, 113–25, 200–5

floating rate bonds 39–40, 51, 54, 92–4, 165–7,

170–3, 200, 203–7, 209–10, 219–21, 268–9

floors 11, 264–9, 273, 286–8

forwards 141–9, 156–8, 162–7

futures 141–9, 156–8, 162–7

hedging 8, 119–21

mortgages 9, 11, 15, 21, 94–6, 118–21

options 244–63, 264–9, 286–94

parallel rates 13, 15, 17, 117, 120–1, 197

present values 23–4, 27–66, 119–21, 171, 206–29

rates/yields contrasts 31–4, 40–1


statistics 225–9

swaps 12–15, 19, 24, 67, 73, 83, 86, 89–94,

199–229, 267–72

term structure 40–9, 94

interest-only MBSs (IOs) 96, 119–21

internal rate of return (IRR), concepts 36–7

International Swaps Dealers Association (ISDA)

19, 67

inverse floaters, concepts 170, 219–21, 291–2

inverted yield curves see downward-sloping . . .

investment banks

concepts 1–2, 17–19, 22, 101–2, 109, 200


investment-grade bonds, concepts 87–8, 124–5

investments


efficiency issues 1–2, 18–19, 234–5

savings 1, 18–19, 108–9, 234–5

speculation 15–16

investors

concepts 1–8, 9, 15–16, 18–19, 20–2, 234–5

institutional traders 8, 11–12, 15–16

risk-sharing activities 2–8, 18–22, 109

types 20–2

IOs see interest-only MBSs

IPSs see index principal swaps

IRR see internal rate of return

ISDA see International Swaps Dealers Association

issuers

bonds 79–90, 200–5

commercial paper 75, 77

concepts 1–2, 9, 75, 77–90, 200–5

Italy 71, 98

Japan 17–18, 20, 64–6, 71, 73, 76–83, 102, 103–6,

109–11, 130, 141, 155–9, 212, 228, 242–3,

262–4, 278–9

bond markets 79–83, 86–7, 278–9

322 Index


equity secondary markets 102, 103–6

financial institutions 17–18

government bonds 79–83, 86–7

JPY 64–6, 71, 130, 155–9, 212, 228, 242–3, 262–4

Tokyo Stock Exchange 19, 104

Jorion, P. 304

JPY 64–6, 71, 130, 155–9, 212, 242–3, 262–4

junk bonds see non-investment-grade bonds

KMV 301–2

knock-ins, concepts 272

knock-outs, concepts 272

lending/borrowing rates

futures 143–9, 175–6, 181–7, 190–8

options 256, 275–6

leverage concepts 9, 87–8, 170, 220–1, 291–2

leveraged buyouts 87–8

leveraged floaters 170

leveraged inverse floaters, concepts 220–1, 291–2

LIBID 142–9, 177

LIBOR see London interbank offered rate

life insurance 3, 6, 10–11

see also insurance . . .

LIFFE 141, 149–50, 153–4, 187

limit orders 105

limited liability companies see companies

linear approximations, duration 54–5

liquid markets 102–12

bid–ask spreads 47

swaps 90

liquidations, companies 59, 298–300

livestock markets 111–12

load-matching behaviour, electricity markets 112

loan extensions, interest-rate futures 145–6

loan markets, concepts 2, 9, 18–19, 142–9, 305–12

local volatility, concepts 275, 286–7

London interbank offered rate (LIBOR)

11–12, 31–2, 78–9, 92–4, 142–51, 157–9,

161–73, 176–87, 191, 200, 204–24, 244, 265–9,

286–8, 304–5

credit derivatives 307

options 244, 265–9, 286–8

swaps 200, 204–24, 267–9

synthetic LIBOR forwards 163–4, 191

London Stock Exchange (LSE) 103–4

long exposures, risk concepts 9–10

long-dated commodity options, concepts

293–4

long-maturity securities, concepts 67

long-volatility positions, options 259–60

LSE see London Stock Exchange

LTCM 22

Macaulay duration 52–6

see also duration

McCauley, R. 82

Marcus, Alan J. 60, 125

mark-to-market settlements, concepts 10, 19,

115–16, 135–41, 207–29, 241–2, 279

market model

concepts 125–8

equities 125–8

market orders, equities 105–6

market players, types 20–2

market risks

concepts 21–2, 60

premiums 60

speculation 21–2

market-neutral hedges 125–9

matched books, repurchase agreements 71–2

mathematics see financial mathematics

MATIF 149–50

maturities

bonds 40–9, 88, 94, 113–16, 167–70

corporate bonds 88

forward-rate agreements 147–9, 163–4, 168–70

options 273, 283, 287–8


swaps 94, 200–29

term structure of interest rates 40–9, 94, 167–70

MBSs see mortgage-backed securities

mean reversion, concepts 262–4, 287

medium-term notes (MTNs), concepts 88

Merton, Robert C. 256

metals 111–12, 159, 161–2, 224–5

Mexico 77–8

Microsoft 285

Milken, Michael 20

mini contracts see unit payoff claims

modified duration

see also duration

concepts 52–6, 116–21

definition 52

examples 52–5, 116–21

money markets 16, 31, 37, 67–79, 142–9

see also commercial paper; Eurocurrencies; Fed

. . .; repurchase agreements; Treasury Bills

concepts 16, 31, 37, 67–79, 142–9

day-count conventions 31, 68–74

definition 67

desks 16

instrument types 67–79

statistics 67–79

Monte Carlo simulations 96

Moody’s 77–8, 87–8, 90, 301–2

Morgan Stanley Capital International 106

mortgage-backed securities (MBSs) 1, 24, 67, 94–6,

118–21, 222–3, 292

see also fixed income . . .

Index 323

concepts 94–6, 118–21, 222–3, 292

definition 94–5

option-adjusted spreads 96, 292

pass-through certificates 95–6, 292

strips 96, 119–21

valuations 96

mortgages

amortizing 38–40

concepts 9, 11, 15, 16, 21, 24, 94–6, 118–21,

222–3, 292

interest rates 9, 11, 15, 21, 94–6, 118–21

negative convexity 118–21

periodic caps 272–3

prepayment risks 16, 94–6, 118–21, 222–3

motor car analogies, financial markets 1, 22–3

MSFT 125–8

MTNs see medium-term notes

multi-dimensional cubes, hedging 8

multiplier concepts

futures 139–49, 159, 168–70

options 243–4

munis

concepts 69–70, 88–90

U.S. 69–70, 88–90

yields 88–90

mutual funds 1, 15, 17, 21–2, 75, 103, 106

Myers, Stewart C. 60, 125

NASD see National Association of Securities

Dealers

NASDAQ 19, 103, 106, 149–53

National Association of Securities Dealers (NASD)

103

NAV see net asset value

negative convexity

concepts 118–21

mortgages 118–21

negotiable CDs 74

net asset value (NAV) 107

Netherlands 86–7, 98, 104

New York Stock Exchange (NYSE) 19, 102–3

Nikkei 106, 149–50, 243–4, 274

no-arbitrage conditions, concepts 15, 23–4, 47,

177–81, 190

non-investment-grade bonds, concepts 87–8, 124–5

non-price variables, options 243–4

Norway 77, 82

notional amounts, swaps 92–4

NQLX 153–4

NYMEX 161–2

NYNEX 174

NYSE see New York Stock Exchange

NZD 64–6

OASs see option-adjusted spreads

off-market swaps

see also swaps

concepts 92–4, 205–7, 210–11

off-the-run T-Bills 84

oil products 111–12, 161–2, 224, 288, 293–4

on-market swaps

see also swaps

concepts 92–4

on-the-run T-Bills 41–2, 83–4

ONE see OneChicago

one-price law, currencies 66

OneChicago (ONE) 153–4

OPEC see Organization of Petroleum Exporting

Countries

open outcry, concepts 137–8

opportunity costs, capital 20–1, 27

opportunity returns, concepts 20–1

optimization software, portfolio strategies 128–9

option-adjusted spreads (OASs) 96, 291–2

concepts 292

mortgage-backed securities 96, 292

options 5–6, 10–11, 14–15, 23–4, 58–9, 106, 225–9,

231–94, 305–12

see also American . . .; European . . .

arbitrage 14–15, 240–2, 261, 275–94

binomial pricing 244–64, 276–94

Black–Scholes model 256–64, 279–85

bonds 265–9, 278–9, 283–92

book-running activities 279–83

borrowing-against-the-box strategies 275–6

borrowing/lending rates 256, 275–6

buy–write strategies 238–40

caps 244, 264–9, 273, 286–8, 291–2

cash-and-carry static arbitrage 275–9

commodities 275–6, 277–8, 288, 293–4

composite payoffs 236–40

concepts 5–6, 10–11, 14–15, 23–4, 58–9, 106,

225–9, 231–94, 305–12

contingent delivery times 10–11, 23–4



correlation risks 288, 294

covered calls 7, 238–40


credit spread options 308–10

currencies 242–3, 261–2, 275–8, 283–9, 290–1

different underlyings 283–9

dividends 257–8

economic variables 294

embedded options 58, 173–4, 228, 289–92,

311–12

energy markets 294

equities 235–94

examples 235–94

exotic options 272–4

324 Index

mortgage-backed securities (cont.)

expiry considerations 235–41, 244–64, 276–94


floors 264–9, 273, 286–8

forward-rate agreements 267–72

forwards 241–2, 251, 267–72, 275–9

gambling elements 233–5, 288, 294

gammas 260

hedging 245–64, 279–94

implied volatility 13, 260–4, 279–94

indexes 10–11, 106

information 12

insurance 233–4, 241, 278–9, 292–4

interest rates 244–63, 264–9, 286–94

mean reversion 262–4, 287

multiplier concepts 243–4

non-price variables 243–4

over-the-counter markets 236

payoff diagrams 235–41

payoffs 233–94

physical settlements 236

portfolios 259–64, 279–94

premiums 233–41, 245–58, 276–94

pricing models 58, 96, 244–64, 276–94

probabilities 245–58

pure arbitrage 12–13, 175–6

quantos 274, 290–1

relative value arbitrage 14–15, 240–2, 261, 275–94

replication strategies 234–5

risk-sharing activities 234–5, 241–2

risk-taking activities 11, 234–5, 241–2

skews 261–4, 275, 282–5

smiles 261–4, 282–3

speculation 233–40, 261, 294

spreads 12, 237–40, 286–8, 291–2


static arbitrage 275–9

statistics 225–9

stock indexes 243–4, 262–4, 277–9, 285

straddles and strangles 236–7

strike prices 235–94

swaps 12–13, 267–73, 287–9, 291–2

swaptions 12–13, 269–73, 287–9

temperature readings 243–4

terminology 235–44

transaction costs 277–8, 283

types 5–8, 10–12, 235–44, 269–74, 278–9, 288–91,

308–10

underlying assets 235–41, 244–64, 276–94

valuations 12, 24, 58, 96, 236, 240–58, 276–94

vegas 259–64, 280–8

volatilities 251, 256–64, 279–94

warrants 278–9, 289, 291

writers 236, 238–40

options markets

concepts 2

definition 2

Orange County 16

ordinary annuities, concepts 29–32

Organization of Petroleum Exporting Countries

(OPEC) 111

original issues segment

see also primary markets


OTC Bulletin Board 103

OTC see over-the-counter markets

out-of-the-money options

see also options

concepts 236, 259–60, 263–4, 267

over-the-counter markets (OTC) 1, 13, 18–20, 31,

67–71, 83, 88, 103, 106, 109–10, 147–9, 173–4,

224–9

see also corporate bonds; government bonds

commodities 111–12, 173–4, 224–9

concepts 1, 13, 18–20, 31, 67–71, 83, 103, 109–10,

173–4, 224–9

currencies 109–10, 225–9

day-count conventions 31, 68–74, 83

forward-rate agreements 147–9, 225–9

options 236

statistics 225–9

trading mechanisms 103, 106

overnight refinancing, banks 70

overview 1–24

P/E ratios see price/earnings . . .

PAC see planned amortization class

pairs trading, concepts 14

‘‘paper’’ 18–19

parallel interest rates 13, 15, 17, 117, 120–1, 197

pass-through certificates, mortgage-backed

securities 95–6, 292

path dependence concepts 6

payment for order flow, concepts 106

payoffs

cash-and-carry replication 14–15, 23–4

composite payoffs 236–40

concepts 2–8, 14–15, 23–4


diagrams 235–41

options 233–94

state-contingent claims 2–8

PEG see price/earnings/growth

pension funds 1, 9, 15, 21–2

percentage basis, concepts 131–2

performance measures, portfolios 125–9

periodic caps, concepts 272–3

perpetuities 61–4

petroleum products 111–12, 161–2, 224, 288,

293–4

Pfizer (PFE) 14

physical settlements, options 236

Index 325

plain vanilla swaps

see also swaps

concepts 91–4, 204–5, 207, 212–29

planned amortization class (PAC), concepts 96

PLZ 130

Poland 130

pork markets 111–12

portfolio duration

concepts 56–8

examples 56

portfolio immunization

see also duration matching

concepts 56–7, 116–21

portfolio strategies

blipping method 196–9

equities 125–9

non-diversified portfolios 126–8

optimization software 128–9

options 259–64, 279–94

performance measures 125–9

returns databases 128–9

swaps 210–11

Portugal 77, 104

POs see principal-only MBSs

PPCs see prospectus prepayment curves

preferred stocks 67

premiums


market risks 60

options 233–41, 245–58, 276–94

prepayment risks

asset-backed securities 97, 222–3

mortgages 16, 94–6, 118–21, 222–3

present values

concepts 23–4, 27–66, 113, 119–21, 171, 206–29,

298–9

interest rates 23–4, 27–66, 119–21, 171, 206–29

price value of a basis point (PVBP)

concepts 58, 197

definition 58

price/earnings ratios (P/E ratios), concepts 63–4

price/earnings/growth (PEG), concepts 64

price/yield relationships, bonds 49–58, 148

pricing theory

futures 150–4

options 58, 96, 244–64, 276–94

swaps 93–4, 199, 211–29

primary markets

concepts 1–2, 17–20, 67, 101–2, 108–9

equities 18–20, 101–2, 108–9

international comparisons 19

regulations 19, 75, 108–9

secondary markets 20, 67

primary risks, secondary risks 15–16, 113

Prime 199–200

principal-only MBSs (POs) 96, 119–21

private placements, concepts 19

probabilities

default risks 298–300

options 245–58

profits

arbitrage concepts 2, 12–16

traders 15

program trades, concepts 108, 182–7

promissory notes 74–9

see also commercial paper

proprietary desks, concepts 16

prospectus prepayment curves (PPCs) 97–9

PSA see public securities association

pseudo arbitrage see relative value arbitrage

public securities association (PSA) 95, 97, 118–21

pure arbitrage

concepts 12–16, 24, 113, 175–6

definition 12–13

examples 12–13

speculation contrasts 15

pure assets, concepts 159–60

put options 10–11, 12, 235–94

see also options

concepts 235–94

put spreads, concepts 237–40

put-protected stock, concepts 239–40

puttable bonds, concepts 291–2

PVBP see price value of a basis point

QEY see quarterly equivalent yield

quantos, concepts 274, 290–1

quarterly equivalent yield (QEY) 32–3

quarterly example, compounding 29–33

quarterly rollover example, compounding 30–3

quasi-arbitrage see relative value arbitrage

quote conventions

currencies 64–5, 109–10, 154–5, 242–3

forward FX rates 154–5

Rajendra, Ganesh 97–8

range bonds, concepts 222

range options, concepts 273–4

rates, concepts 31–4, 40–1

ratings

agencies 67, 77–8, 87–90, 124–5, 299–302

bonds 40–1, 59, 77–8, 87–90, 124–5, 299–302

debt 40–1, 59, 67, 77–8, 87–90, 124–5, 200,

299–302

realized volatility, implied volatility 262

rectangular chessboards, state-contingent claims

3–8

redundant securities 152, 155–6, 234–5

reference securities, credit derivatives 306–12

registration requirements, bonds 34

326 Index

regulations

primary markets 19, 75, 108–9


relationships, arbitrage 2, 162–7, 175–6,

207–29, 311–12

relative price, risk 2

relative value arbitrage

commodities 131–2

concepts 12–16, 21–2, 113–32, 175–98, 275–94,

311–12

currencies 129–30, 176–81

definition 13

equities 125–9, 182–7

examples 13–16, 113–32, 175–98

fixed income securities 113–32

options 14–15, 240–2, 261, 275–94

speculation contrasts 14–16, 21–2, 294, 311–12

spot markets 113–32, 176–81

yield curves 113–32

Remolona, E. 82, 89, 94

replication strategies

coupon 113–16, 162–7

options 234–5

stock index futures 151–4, 175–6

synthetic replications 45–7, 85, 162–7, 175–6,

189–98, 241–2, 275–6

repurchase agreements (RPs) 71–2, 85, 171–3,

187–92


concepts 71–2, 85, 171–3, 187–92

forwards 171–3, 187–92

reverse repos, concepts 71–2

special repos 171–3

resale segment

see also secondary markets


retail price index (RPI) 86

retained earnings, concepts 63–4

return on investment (ROI), concepts 62–4

returns 2–8, 15–16, 20–1, 62–4, 125–9

databases 128–9

factor-based models 125–8

Fama–French model 126–7

opportunity returns 20–1

portfolio performance 125–9

risk 2–8, 15–16, 20–1, 62–4

Reuters 109, 154–5

reverse cash-and-carry, concepts 135–6, 152, 160,

187

reverse repos, concepts 71–2

risk

see also interest-rate risk

arbitrage concepts 2, 12–16

basis risks 294

concepts 2–10, 20–2, 49–58, 217–18, 225–9

credit risks 15, 20, 23–4, 94, 113, 137–41, 147, 276,

295–312

diversification strategies 9–10, 17, 126–8, 293

‘‘good/bad’’ distinctions 9–10

path dependence concepts 6

relative price 2

returns 2–8, 15–16, 20–1, 62–4

spread risks 15–16, 113, 121–5, 217–18

statistics 225–9

swaps 211, 217–18, 225–9

types 15–16, 20–2, 23–4, 49–58, 217–18, 294

value-at-risk 304

volatilities 251, 256–64

risk arbitrage see relative value arbitrage

risk-free government bonds

credit spreads 297–8

hedging 8

risk-neutral probabilities, options 245–58

risk-sharing activities

concepts 2–11, 18–22, 109, 137–8, 234–5, 305–12


definition 7

examples 7–11, 21–2, 234–5

forwards/futures 137–8

hedgers 8, 21–2

options 234–5, 241–2

risk-taking activities, options 11, 234–5, 241–2

Robinson, Franklin L. 31

ROI see return on investment

Roll, R. 258

RPs see repurchase agreements

Rubinstein, Mark 244

Rule 144-A 19

Russell 2000 index 149

S&P 100 10, 106, 285

S&P 500 107, 125–8, 137–8, 149–54, 182–7, 223–4,

262–4, 282–3

S&P see Standard and Poor’s

SAEY see semi-annual equivalent yield

Sallie Mae see Student Loan Marketing Association

Saudi Arabia 111

savings

efficiency issues 1–2, 18–19, 108–9, 234–5

investments 1–2, 18–19, 108–9, 234–5

productive ventures 1, 18–19, 108–9, 234–5

Scholes, Myron 256–64

SEAQ 104

seasoned equity offerings (SEOs) 102, 108–9

SEC see Securities and Exchange Commission

secondary markets

concepts 1–2, 17–20, 67, 75, 90, 101–9


developing countries 19–20

economic roles 108–9, 234

electronic communications networks 103, 105

Index 327

equities 101–9

government bonds 86–7

international comparisons 19, 75, 101–9

London Stock Exchange 103–4

NASDAQ 19, 103, 106

New York Stock Exchange 19, 102–3

primary markets 20, 67

regulations 108–9

Tokyo Stock Exchange 19, 104

types 101–9

secondary risks 15–16, 113

Securities and Exchange Commission (SEC) 19,

106

securities markets

see also capital . . .; money . . .

concepts 67

securitization

concepts 95–9

stages 95–6

trends 18

sell-side participants

see also hedging

concepts 21–2

semi-annual equivalent yield (SAEY) 32–6, 42,

167–8

SEOs see seasoned equity offerings

sequential pay CMOs, concepts 95–6

serial bonds, U.S. munis 90

SETS 104

short exposures, risk concepts 9–10

short-term munis, U.S. 69–70

short-volatility positions, options 259–60

Siegel, Daniel R. 188

Siegel, Diane F. 188

SIMEX 141–2

Singapore 141–2

single stock futures, concepts 153–4

skews, volatilities 261–4, 275, 282–5

smiles, volatilities 261–4, 282–3

‘‘soft dollar’’ payments 105–6

Spain 87, 98, 104

Sparkasse 20

special purpose vehicles (SPVs) 97

special repos, concepts 171–3

specialists, New York Stock Exchange 102–3

speculation

arbitrage contrasts 14–16, 21–2, 294, 311–12

concepts 2, 8, 9–10, 13–16, 20–2, 24, 109–10,

122–3, 131–2, 204–5, 233–40, 294, 311–12

definition 21

examples 14–16, 20–2, 122–3, 204–5

hedging contrasts 20–2, 24, 311–12

options 233–40, 261, 294

pure arbitrage contrasts 15

relative value arbitrage contrasts 14–16, 21–2,

294, 311–12

spot markets 2, 3–4, 6, 10–11, 23–4, 25–132, 175–98

complexity issues 11–12

concepts 2, 3–4, 6, 10–11, 23–4, 113–32

definition 2

financial mathematics 27–66, 128–9

forward contrasts 10

information 11–12

relative value arbitrage 113–32, 176–87

valuations 12

spot rates

see also zero-coupon rates

concepts 32

spot–forward arbitrage

bond futures 187–92

cash-and-carry 155–6, 160, 175–98

concepts 162–7, 175–98

fixed income securities 187–92

hedging 193–8

interest rates 162–7


stock indexes 108, 182–7

spread arbitrage, concepts 121–3

spread options 12, 288, 294

spread risks, concepts 15–16, 113, 121–5,

217–18

spread strategies, corporate bonds 121–3

spreads


options 12, 237–40, 286–8, 291–2

swaps 94, 217–18

SPVs see special purpose vehicles

SPX 186–7, 223–4

stacking concepts 293

standard deviations

Black–Scholes model 256–64

credit risks 303–5

Standard and Poor’s (S&P) 10, 77, 87–8, 90, 106–7,

125–8, 137–8, 149–54, 182–7, 223–4, 262–4,

282–3, 285, 301–2

state variables, concepts 265

state-contingent claims

see also valuations

chessboards 3–8, 193

concepts 2–12

examples 3–11

payoffs 2–8

static arbitrage

concepts 12–16, 23–4, 115–16, 275–9

options 275–9

static relative arbitrage, concepts 13–16, 115–16

step-up coupons, concepts 87–8

Stigum, Marcia 31

stock baskets 15, 24, 184–7, 279–89

stock exchanges

328 Index

secondary markets (cont.)

see also financial markets; primary . . .;

secondary . . .

background 18, 102–9

economic roles 108–9, 234

trading mechanisms 102–6

stock index futures 15, 137–8, 149–54, 175–6, 182–7,

223–4, 262–4, 277–9

concepts 15, 137–8, 149–54, 175–6, 182–7, 223–4,

262–4, 277–9

dividends 152–4, 182–7

examples 149–54, 182–7, 223–4, 262–4

lock-in prices 150–1

replication strategies 151–4, 175–6

stock indexes 15, 106–7, 125–9, 137–8, 149–54,

175–6, 182–7, 223–4, 243–4, 262–4, 277–9, 285

arbitrage 182–7, 277–9, 285

capitalization-weighted indexes 106–7

concepts 15, 106–7, 125–9, 137–8, 149–54, 175–6,

182–7, 223–4, 243–4

FTSE 100 104, 106, 149–50, 274

Nikkei 106, 149–50, 243–4, 274

options 243–4, 262–4, 277–9, 285

S&P 100 10, 285

S&P 500 107, 125–8, 137–8, 149–54, 182–7,

223–4, 262–4, 282–3

spot–forward arbitrage 108, 182–7

tracking risks 186–7, 288

stocks 1, 6, 58–64

see also equities

straddles, concepts 236–7

strangles, concepts 236–7

strike prices

options 235–94

smiles 261–4, 282–3

strips

concepts 45–9, 83–5, 96, 113–16, 119–21, 162–7,

176, 215–16

mortgage-backed securities 96, 119–21

structured finance

callable bonds 221–2, 289–92, 310

concepts 218–23, 289, 291–2, 305–12

swaps 218–23

student loan ABSs 96–8

Student Loan Marketing Association (Sallie Mae)

69

subordinated bonds, concepts 87–8

subtraction concepts, valuations 6

swaplets 148, 199

swaps 12–15, 19, 24, 67, 73, 83, 86, 89–94, 109–10,

148, 199–229, 267–72, 287–9, 291–2, 306–7, 310

see also Eurocurrencies; fixed income . . .

applications 199–229

arbitrage 94, 207–29, 310

assignments 90

book-running activities 210–17

commodities 224–5

concepts 12–15, 19, 24, 67, 73, 83, 86, 89–94,

109–10, 148, 199–229, 267–72, 287–9, 291–2,

306–7, 310

counterparty default 94, 306–10

credit default swaps 306–10

currencies 92–3, 110, 200–3, 267–72

definition 91–2, 148

equities 223–9

examples 200–29

fixed-for-fixed currency swaps 200–3,

205–7, 208–10

fixed-for-floating interest-rate swaps 203–7,

209–10, 222–3

forwards 199, 207–29, 270–2

government bonds 90–4, 211, 217–18

hedging 211–29

index principal swaps 222–3

innovations 224–5

interest rates 12–15, 19, 24, 67, 73, 83, 86, 89–94,

199–229, 267–72

international comparisons 89–94, 200–29

inverse floaters 219–21, 291–2

liquidity issues 90

maturities 94, 200–29

off-market swaps 92–4, 205–7, 210–11

options 12–13, 267–73, 287–9, 291–2

plain vanillas 91–4, 204–5, 207, 212–29

portfolio strategies 210–11

pricing 93–4, 199, 211–29

risk 211, 217–18, 225–9

settlement conventions 92–3, 148

spreads 94, 217–18


structured finance 218–23

term structure of interest rates 94

total-rate-of-return swaps 307

types 91–4, 199, 204–10, 218–25, 306–7

U.K. 200–29, 310

U.S. 91–4, 200–29

valuations 93–4, 199, 211–29

swaptions, concepts 12–13, 269–73, 287–9

Sweden 76, 78, 86–7, 98

SWIFT 64, 109

Switzerland 82, 86–7, 109–10, 130, 212, 262

synthetic forwards, concepts 163–7, 175–98, 241–2,

275–6

synthetic LIBOR forwards, concepts 163–4, 175–6,

191

synthetic replications 45–7, 85, 162–7, 175–6, 241–2

synthetic zeros, concepts 164–5, 189–98

T-Bills see Treasury Bills

tap systems, bond issues 86–7

taxation, U.S. munis 69–70, 88–90

TED spread, concepts 74

temperature readings

Index 329

futures 136–7

options 243–4

term repos, concepts 71–2

term structure of discount rates, concepts 41–2

term structure of interest rates

see also yield curves

arbitrage arguments 45–9

bootstrapping 44–9, 167–70, 189–93

concepts 40–9, 94, 288

economic cycles 42

examples 42–9, 167–70, 227–8

swaps 94, 211–12, 227–8

types 41–2

yield to maturity 43–4

zero-coupon rates 42–9, 167–70, 189–98, 211–29

term structure of par rates, concepts 41–2

Tiger 22

time considerations


valuations 6

time to maturity, price/yield relationships

51–8

time value of money 10, 27

concepts 27

TIPS see Treasury inflation-protected securities

tobacco companies 12

Tokyo Stock Exchange (TSE) 19, 104

TOPIX 106

total-rate-of-return swaps, concepts 307

tracking risks, concepts 186–7, 288, 294

traders, profit sources 15

trading desks 16

trading floors, futures 137–8

tranches, collateralized mortgage securities 95–6

transaction costs

concepts 66, 115–16, 129–30, 176, 193, 277–8, 283

currencies 66, 129–30

options 277–8, 283

Treasury Bills (T-Bills)


concepts 41–2, 68–9, 73–4, 113–16, 286

on-the-run T-Bills 41–2, 83–4

TED spread 74

U.S. 20, 33, 41–2, 68–9, 73–4, 83–4, 113–16, 286

Treasury curves, concepts 85

Treasury inflation-protected securities (TIPS) 85

Treasury notes, U.S. 41–2, 83–5, 113–16, 141–9

tree simulations 96, 244–64, 276–94

triangular arbitrage, concepts 15, 110–11, 129–30

TSE see Tokyo Stock Exchange

Turkey 82

twos–tens trades, spot relative value trades 117–21

U.K. 22, 64–6, 71, 76–7, 86–7, 93, 103–6, 109–11,

141–9, 200–29, 262–4, 303–5, 310

see also Europe


Bank of England 86–7

currencies 64–6, 71, 93, 109–11, 200–29, 262–4,

303–5, 310

day-count conventions 32, 204–5, 217–18


Eurocurrencies 72–4, 142–9

Financial Services Authority 106

futures 141–9, 153–4

GBP 64–6, 71, 93, 109–11, 200–29, 262–4, 303–5,

310

government bonds 86–7, 310

index-linked gilts 86–7

London interbank offered rate 11–12, 31–2, 78–9,

92–4, 142–51, 157–9, 161–73, 176–87, 191, 200,

204–24, 244, 265–9, 286–8, 304–5

London Stock Exchange 103–4

swaps 200–29, 310

U.S. 11, 17–20, 22, 32–6, 41–2, 64–6, 68–90, 93,

102–6, 109–11, 130, 141, 154–9, 167–70,

176–81, 187–92, 200–29, 242–3, 273–4, 303–5


bonds 32–6, 41–2, 79–90, 167–70, 187–92


corporate bonds 79–83, 87–90

currencies 64–6, 71, 93, 109–11, 130, 141, 154–9,

176–81, 200–29, 242–3, 273–4, 303–5

day-count conventions 32, 36, 68–9, 83, 113–16,

168–70, 204–5, 217–18

debt-outstanding statistics 83–5

electricity markets 173–4, 294


Eurodollars 31, 72–4, 141–9, 167–70, 191–8,

212–29

Fed Funds 70, 92, 286

federal agency discount notes 69, 85

Federal Reserve 67–9, 261–2

financial institutions 17–18, 70

forwards 136–74, 176–98

futures 136–74, 187–92

government bonds 20, 33, 41–2, 68–9, 73–4,

79–86, 113–16, 141–9, 286

munis 69–70, 88–90

New York Stock Exchange 19, 102–3

primary markets 19

repurchase agreements 71–2, 85

Securities and Exchange Commission 19, 106

short-term munis 69–70

swaps 91–4, 200–29

Treasury Bills 20, 33, 41–2, 68–9, 73–4, 83–4,

113–16, 286

Treasury notes 41–2, 83–5, 113–16, 141–9

USD 64–6, 71, 93, 109–11, 130, 141, 154–9,

176–81, 200–29, 242–3, 273–4, 303–5

UBS 154–8

330 Index

underlying assets, option concepts 235–41, 244–64,

276–94

underwriting practices, equities 101–2

unemployment rates, options 294

unit payoff claims


concepts 2–8

unit trusts 1, 17

universal banks

see also financial institutions

concepts 17–18

unwinds, swaps 90

upstairs market, trading mechanisms 105

upward-sloping yield curves 42, 118–21, 228–9

USD 64–6, 71, 93, 109–11, 130, 141, 154–9, 176–81,

200–29, 242–3, 273–4, 303–5

valuations

bonds 49–58, 170–3

companies 59

concepts 6, 11–12, 24, 58, 59–64, 96, 135–6,

155–6, 160, 170–3, 177–87, 236, 240–58

convertible bonds 289–90

equities 60–4, 101–9

floating rate bonds 170–3

forwards 12, 135–6, 155–6, 160, 177–87

futures 135–6, 150–4, 160, 177–87

good models 6

information 11–12

mortgage-backed securities 96

options 12, 24, 58, 96, 236, 240–58, 276–94

spot markets 12

state-contingent claims 6

subtraction concepts 6

swaps 93–4, 199, 211–29

time considerations 6

zero-valuations 6

value-at-risk 304

Vanguard Equity Index 107

variation margins, concepts 135–41, 150–4

vegas, concepts 259–64, 280–8

venture capital 109

volatilities

arbitrage 279–94

caps 286–8

concepts 256–64, 279–94

historical volatility 260–1, 293

implied volatility 13, 260–4, 279–94

local volatility 275, 286–7

long-term/short-term contrasts 286–7

options 251, 256–64, 279–94

skews 261–4, 275, 282–5

smiles 261–4, 282–3

Wall Street Journal 68, 83, 138–43, 150, 154, 161–2,

168, 176, 212

warrants, concepts 278–9, 289, 291

weighted average time to cash flows, duration 52–6

when issued securities (WI) 84

Xetra 104

yield curves

see also term structure of interest rates

concepts 15, 23–4, 40–9, 94, 113–32, 167–70,

195–8, 211–12, 227–9, 288


curve-blipping method 195–9

examples 42–9, 94, 113–32, 167–70, 227–8

forwards 167–73, 189–98, 211–29

relative value arbitrage 113–32

shapes 42, 94, 113–32, 167–70, 227–9, 288

spread options 288

swaps 94, 211–12, 227–9

tilts 15, 42, 113–32

yield to maturity (YTM)

concepts 35–8, 43–8, 49–58, 84–5, 114–16, 218,

297–8, 302, 308–10

examples 36–8, 43–4, 49–58, 84–5, 114–16, 218

parallel changes 56, 117, 120–1

price/yield relationships 49–58

term structure of interest rates 43–4

zero-coupon rates 37–8, 49–58, 84–5, 114–16

yields

concepts 31–4, 40–1, 49–58

convexity 49–58, 116–25, 148, 213, 273

modified duration 52–6, 116–21

munis 88–90

parallel changes 56, 117, 120–1, 197

price value of a basis point 58, 197


YTM see yield to maturity

Z bonds

see also mortgage-backed securities

concepts 96

zero bootstrap, concepts 44–5, 167–70

zero-coupon rates

add-on/discount basis 33

concepts 27–8, 32–49, 84–5, 113–25, 162–70,

189–98, 200, 211–29

duration 54

forwards 164–70, 189–98, 211–29

graphical representation 32–5, 167–70

interest-rate risk 51–2

synthetic zeros 164–5, 189–98

term structure of interest rates 42–9, 167–70,

189–98

yield to maturity 37–8, 49–58, 84–5, 114–16

zero-coupon stripping, concepts 113–16, 162–7

Index 331

Economy & Finance

An arbitrage guide to financial markets